A  KINETIC  THEORY 


OF 


GASES  AND  LIQUIDS 


BY 

RICHARD   D.   KLEEMAN, 

D.Sc.  (Adelaide),  B.A.  (Cantab.) 

Associate  Professor  of  Physics,  Union  College;   Consulting  Physicist 

and  Physical  Chemist;  formerly  1851  Exhibition  Scholar  of  the 

University  of  Adelaide,  Research  Student  of  Emmanuel 

College,    Cambridge,    Mackinnon    Student,  of   the 

Royal  Society  of  London,  and  Clerk  Maxwell 

Student  of  the  University  of  Cambridge 


FIRST    EDITION 


NEW  YORK 

JOHN   WILEY   &   SONS,   INC, 

LONDON:  CHAPMAN  &  HALL,  LIMITED 
1920 


K  :: 


Copyright,  1920 

BY 

RICHARD  D.  KLEEMAN 


BRAUNrtOHTH    &    CO. 

BOOK    MANUFACTURERS 

BROOKLYN.    N.    Y. 


DEDICATED 

TO    MY    PARENTS 


PREFACE 


THE  object  of  writing  this  book  is  to  formulate  a  Kinetic 
Theory  of  certain  properties  of  matter,  which  shall  apply 
equally  well  to  matter  in  any  state.  The  desirability  of 
such  a  development  need  not  be  emphasized.  The  difficulty 
hitherto  experienced  in  applying  the  results  obtained  in  the 
case  of  the  Kinetic  Theory  of  Gases  in  the  well-known  form 
to  liquids  and  intermediary  states  of  matter  has  been  pri- 
marily due  to  the  difficulty  of  properly  interpretating  molec- 
ular interaction.  In  the  case  of  gases  this  difficulty  is  in 
most  part  overcome  by  the  introduction  of  the  assumption 
that  a  molecule  consists  of  a  perfectly  elastic  sphere  not 
surrounded  by  any  field  of  force.  But  since  such  a  state  of 
affairs  does  not  exist,  the  results  obtained  in  the  case  of  gases 
hold  only  in  a  general  way,  and  the  numerical  constants 
involved  are  therefore  of  an  indefinite  nature,  while  in  the 
case  of  dense  gases  and  liquids  this  procedure  does  not  lead 
to  anything  that  is  of  use  in  explaining  the  facts. 

Instead  of  an  atom,  or  molecule,  consisting  of  a  per- 
fectly elastic  sphere,  it  is  more  likely  that  each  may  be 
regarded  simply  as  a  center  of  forces  of  attraction  and 
'repulsion.  If  the  exact  nature  of  the  field  of  force  sur- 
rounding atoms  and  molecules  were  known,  it  would  be  a 
definite  mathematical  problem  to  determine  the  resulting 
properties  of  matter.  But  our  knowledge  in  this  connection 
is  at  present  not  sufficiently  extensive  to  permit  a  develop- 
ment of  the  subject  along  these  lines.  But  in  whatever 


viii  PREFACE 

With  the  foregoing  results  as  a  basis,  and  the  modified 
definitions  of  the  free  paths  of  a  molecule  mentioned,  the 
foundation  of  a  general  Kinetic  Theory  can  be  laid  which 
applies  to  matter  in  any  state,  and  which  furnishes  a  num- 
ber of  important  formulae.  These  formulas  may  be  given 
important  extended  forms  containing  quantities  which  are 
arbitrary  in  so  far  as  they  satisfy  the  formulae  as  a  whole. 
Formulae  may  also  be  deduced  along  the  same  lines  involving 
instead  of  the  molecular  free  path  the  projection  of  the  mo- 
tion of  a  molecule  along  a  line.  This  projection  and  its 
period  are  also  arbitrary  in  so  far  as  they  satisfy  the  formulae 
as  a  whole.  Some  of  them  find  an  interesting  application 
in  connection  with  colloidal  solutions.  The  foundation  of 
the  subject  may  be  said  to  be  fairly  complete,  since  it  fur- 
nishes the  structure  about  which  further  advances  may 
be  made  so  that  the  subject  can  be  rendered  more  or  less 
complete.  These  advances  will  consist  largely  in  expressing 
the  constants  involved  in  the  fundamental  formulae  in  terms 
of  the  constants  of  the  molecular  forces  and  the  molecular 
volume.  If  the  formulae  obtained  in  connection  with  vis- 
cosity, conduction  of  heat,  and  diffusion,  be  applied  to  a 
perfect  gas,  they  assume  the  well-known  forms,  but  the  sym- 
bols have  somewhat  different  meanings. 

The  development  presented  is  perfectly  sound  without 
involving  difficult  mathematics.  This  has,  in  fact,  been 
one  of  the  main  objects  kept  in  view.  As  a  physicist  I  have 
often  failed  to  see  the  usefulness  from  a  physical  standpoint 
of  the  extremely  intricate  mathematical  investigations  pur- 
porting to  work  out  to  the  utmost  limit  the  results  of  certain 
assumptions  (usually  in  connection  with  molecular  collision) , 
and  which  have  usually  led  to  results  whose  usefulness  seems 
incommensurate  with  the  labor  involved,  seeing  that  the 
assumptions  are  usually  not  likely  to  be  true.  It  is  desirable 
that  there  should  be  a  simple  and  clearcut  connection  between 


PREFACE  IX 

theory  and  experiment,  and  the  physical  side  of  the  sub- 
ject has  therefore  been  kept  in  prominence. 

The  development  of  a  general  Kinetic  Theory  of  matter 
will  be  of  service  in  the  study  of  chemical  action,  principally 
in  connection  with  the  constant  of  mass-action  and  the  reac- 
tion velocity  constants.  On  the  whole  the  kinetic  aspect 
of  the  chemical  interaction  of  molecules  apart  from  the 
thermodynamical  aspect,  which  does  not  take  into  account 
the  individual  nature  of  the  different  effects  producing  a 
resultant  whole,  can  only  proceed  along  lines  having  such  a 
Kinetic  Theory  as  a  basis.  Thus  it  will  readily  be  seen 
that  the  rapidity  of  a  chemical  reaction  in  a  gaseous  or 
liquid  mixture  must  be  intimately  connected  with  the  num- 
ber of  molecules  crossing  a  square  centimeter  per  second 
in  the  case  of  each  constituent.  The  constant  of  mass-action 
must  evidently  be  intimately  connected  with  the  free  dif- 
fusion path  of  a  molecule,  etc. 

The  study  of  viscosity,  conduction  of  heat,  diffusion, 
etc.,  has  usually  been  confined  to  pure  substances  and  to 
mixtures  whose  constituents  do  not  interact  chemically. 
It  would  evidently  be  of  great  interest  to  study  these  effects 
in  the  case  of  substances  partly  dissociated,  since  the  inter- 
acting between  molecules  is  then  influenced  by  chemical 
affinity.  Further  light  may  be  thrown  on  the  nature  of  this 
property  by  the  application  of  the  formulae  obtained. 

I  have  previously  published  some  investigations  in 
various  scientific  journals  along  the  lines  pointed  out.  Since 
the  development  of  a  subject  can  only  be  gradual,  it  was 
necessary  to  modify  some  of  these  results  before  incorporat- 
ing them  into  this  book.  The  other  investigations  in  the 
book  which  more  or  less  complete  the  Kinetic  Theory  along 
the  lines  mentioned  I  have  not  published  previously.  It 
seemed  that  it  would  be  better  to  present  the  subject  as  a 
whole  to  the  scientific  public,  since  the  various  results  are 


X  PREFACE 

intimately  connected  and  could  therefore  not  be  presented 
in  detached  parts  without  a  good  deal  of  reference  to,  and 
recapitulation  of,  preceding  parts  being  necessary,  to  the 
inconvenience  of  the  reader. 

The  book  has  also  been  brought  up  to  date  in  matters 
not  connected  with  molecular  collision,  and  has  been  treated 
in  a  way  so  that  the  results  are  connected  as  directly  as  pos- 
sible with  the  results  of  experiment. 

The  properties  of  matter  treated  in  this  book  may  be 
said  to  depend  mainly  on  the  dynamical  properties  of  mole- 
cules modified  in  most  cases  by  the  molecular  forces  of  attrac- 
tion and  repulsion.  There  is  evidently,  therefore,  another 
side  to  the  subject  of  the  properties  of  matter,  namely  that 
which  deals  with  those  properties  which  depend  in  the  main 
on  the  molecular  forces  modified  in  some  cases  by  the  dynam- 
ical properties  of  the  molecules.  Thus,  for  example,  the 
internal  heat  of  evaporation  and  the  intrinsic  pressure 
probably  do  not  depend  directly  on  molecular  motion. 
This  part  of  the  physico-chemical  properties  of  matter  will 
be  dealt  with  in  a  separate  book  under  the  title  "  Molecular 
Forces. "  The  present  book  may  serve  together  with  the  fore- 
going as  an  introduction  to  the  study  of  the  purely  ther- 
modynamical  aspect  of  material  properties. 

R.  D.  KLEEMAN. 
UNION  COLLEGE,  SCHENECTADY. 
March,  1919. 


CONTENTS 


CHAPTER  I 

THE    MOLECULAR    CONSTANTS,    AND    THE    DYNAMICAL    PROPERTIES 
OF  A  MOLECULE  IN  THE  GASEOUS  STATE 

SEC.  PAGE 

1.  A  Brief  Historical  Summary  of  the  Development  of  the  Kinetic 
Theory  of  Explaining  the  External  Pressure  and  other  Properties 

of  Gases 1 

2.  A  Direct  Experimental  Proof  that  Matter  consists  of  a  large 
number  of  Entities  or  Atoms 3 

3.  The  Absolute  Mass  of  an  Atom  determined  from  a  Knowledge  of 
the  Electric  Charge  e  carried  by  an  Electron 5 

4.  Indirect  Experimental  Evidence  that  the  Molecules    of  Gases 
and  Liquids  are  in  Rapid  Motion 8 

5.  The  Absolute  Temperature;  and  the  Equation  of  a  Perfect  Gas, 
and  of  a  Mixture  of  Gases  11 

6.  The  Velocity  of   Translation   of  a   Molecule   in  a   Gas  from 
Dynamics 15 

.  7.  Maxwell's  Law  of  Distribution  of  the  Velocities  of  the  Mole- 
cules of  a  Gas 19 

8.  The  Average  Kinetic  Energy  Velocity  of  a  Molecule,  and  the 
Relation  between  the  Kinetic  Energy  of  a  Molecule  and  its 

Absolute  Temperature 27 

9.  The  Equipartition  of  Kinetic  Energy  between  the  Different 
Molecules  in  a  Mixture  of  Gases 31 

10.  The  Number  of  Molecules  per  Cubic  Cm.  in  a  Gas 33 

11.  The  Number  of  Molecules  Crossing  a  Square  Cm.  in  all  Direc- 
tions from  one  Side  to  the  Other  in  a  Gas 34 

12.  The  First  Law  of  Thermodynamics 37 

y  13.  The  Specific  Heats  of  Gases  and  Liquids 39 

14.  Evidence  that  Molecules  and  Atoms  are  surrounded  by  Fields 

of  Force 45 

Molecular  Interaction 48 

xi 


4 


xii  CONTENTS 


CHAPTER  II 

THE    EFFECT    OF    THE    MOLECULAR    FORCES    ON    THE    DYNAMICAL 
PROPERTIES  OF  A  MOLECULE  IN  A  DENSE  GAS  OR  LIQUID 

SEC.  PAGE 

16.  The  Velocity  of  Translation  of  a  Molecule  in  a  Liquid  or 
Dense  Gas  when  passing  through  a  Point  at  which  the  Forces 

of  the  Surrounding  Molecules  neutralize  each  other 51 

17.  The  Total  Average  Velocity  of  Translation  of  a  Molecule  in  a 
Substance 54 

18.  The  Number  of  Molecules  crossing  a  Square  Cm.  in  a  Sub- 
stance of  any  Density  from  one  Side  to  the  Other  per  Second . .     56 

19.  The  Effect  of  Molecular  Volume  on  p  and  n  in  the  Case  of  an 
Imperfect  Gas 57 

20.  The  Expansion  Pressure  exerted  by  the  Molecules  of  a  Sub- 
stance of  any  Density 62 

21.  The  Intrinsic  Pressure;    and  the  Equation  of  Equilibrium 

of  a  Substance 69 

22.  Superior  and  Inferior  Limits  of  n  of  a  Substance 74 

23.  Superior  and  Inferior  Limits  of  Vt  of  a  Substance 75 

24.  The  Real  and  Apparent  Volumes  of  a  Molecule,  and  their 
Superior  Limits 77 

25.  Inferior  Limits  of  n  and  Vt 80 

26.  The  Equation  of  State  of  a  Substance 81 

27.  The  Conditions  that  the  Equation  of  State  has  to  satisfy 86 

28.  The  Relation  of  Corresponding  States 91 

29.  The  Determination  of  the  Quantities  n,  Vt,  and  6,  of  a  Sub- 
stance    95 

30.  An  Equation  Connecting  the  Intrinsic  Pressure,  Specific  Heat, 
and  other  Quantities 103 

31.  The  Mean  Free  Path  of  a  Molecule  under  Given  Conditions.  .  106 


CHAPTER  III 

QUANTITIES  WHICH  DEPEND  DIRECTLY  ON  THE  NATURE  OF 
MOLECULAR  MOTION 

32.  The  Coefficient  of  Viscosity  of  a  Substance 110 

33.  The  Viscosity  Mean  Momentum  Transfer  Distance  of  a  Mole- 
cule in  a  Substance 113 

34.  Formulae  for  the  Viscosity  in  Terms  of  Other  Quantities 116 

35.  Formulae  for  the  Viscosity  of  Mixtures 142 


CONTENTS  xiii 

BEC.  PAGE 

36.  The  Coefficient  of  Conduction  of  Heat 147 

37.  The  Mean  Heat  Transfer  Distance  of  a  Molecule  in  a  Sub- 
stance     148 

38.  Formulae  for  the  Coefficient  of  Conduction  of  Heat  in  Terms 

of  Other  Quantities 150 

39.  Formulae  for  the  Coefficient  of  Conduction  of  Heat  of  a  Mixture 

of  Substances 183 

40.  The  Coefficient  of  Diffusion  of  a  Substance 168 

41.  The  Mean  Diffusion  Path  of  a  Molecule  in  a  Mixture .  .  169 

42.  Formulae  Expressing    the  Coefficient  of  Diffusion  in  Terms 

of  Other  Quantities 170 

43.  Maxwell's  Expression  for  the  Coefficient  of  Diffusion  of  Gases  187 


CHAPTER  IV 

MISCELLANEOUS    APPLICATIONS,     CONNECTIONS,     AND    EXTENSIONS 
OF  THE  RESULTS  OF  THE  PREVIOUS  CHAPTERS 

44.  The  Direct  Observation  of  some  of  the  Quantities  depending 
on  the  Nature  of  the  Motion  of  a  Molecule  in  a  Substance, 
and  their  Use 189 

45.  The   Coefficient   of   Diffusion   in   Connection   with   Osmotic 
Pressure  and  the  Coefficient  of  Mobility 196 

46.  Partial  Intrinsic  Pressures 202 

47.  Conditions  of  the  Equilibrium  of  a  Heterogeneous  Mixture 
such  as  Two  Phases  in  Contact 206 

48.  Osmotic  Pressure  expressed  in  Terms  of  the  Kinetic  Properties 

of  Molecules 208 

49.  The  Velocity  of  Translation  of  Particles  undergoing  Brownian 
Motion 210 

50.  The  Osmotic  Pressure  of  a  Dilute  Solution  of  Molecules  and 
their  Velocity  of  Translation 213 

51.  A  Direct  Determination  of  N,  the  Number  of  Molecules  in  a 
Gram  Molecule 217 

52.  Formulae  involving  Stokes'  Law 220 

53.  Extended  Forms  of  the  Diffusion  Equations 222 

54.  Extended  Forms  of  the  Viscosity  Equations 230 

55.  Extended  Forms  of  the  Heat  Conduction  Equations 238 

56.  The  Constant  Period  Displacement  Diffusion  Equation  and 

its  Applications 242 

57.  The  Constant  Displacement  Diffusion  Equation 252 


xiv  CONTENTS 

SEC.  PAGE 

58.  The  Constant  Period  and  Constant  Displacement  Viscosity 
Equations 255 

59.  The  Constant  Period  and  Constant  Displacement  Heat  Con- 
duction Equations 259 

60.  Another  Method  of  Determining  the  Total  Average  Velocity  of 
Translation  of  a  Colloidal  Particle 262 

61.  The  Distribution  of  the  Molecular  Velocities  in  a  Substance 
not  Obeying  the  Gas  Laws 265 


NOTE 

IT  will  be  useful  to  give  a  list  of  the  most  important 
symbols  used  in  a  general  way  in  this  book,  the  kind  of 
molecule  and  phase  to  which  a  symbol  refers  in  any  given 
case  being  indicated  by  the  addition  of  suffixes  and  dashes 
when  necessary. 

p  =  external  pressure 
Pn  =  intrinsic  pressure 
Ps = Osmotic  pressure 

v  =  volume  of  a  gram  molecule  of  a  substance 
d  =  external  molecular  volume   connected  with  the  con- 
centration changes  of  a  mixture 
6  =  apparent   internal   molecular  volume   connected   with 

molecular  motion 
p  =  density 

T= absolute  temperature 
S  =  specific  heat 
n  =  number  of  molecules  crossing  a  square  cm.  from  one 

side  to  the  other  per  second 
m  =  relative  molecular  weight 
ma= absolute  molecular  weight 
N  =  number  of  molecules  in  a  gram  molecule 
N  with  suffixes  =  molecular  concentration 
V  =  average  kinetic  energy  velocity  of  a  molecule  in  the 

gaseous  state 

Va  =  average  velocity  of  a  molecule  in  the  gaseous  state 
Vt  —  total  average  velocity  of  a  molecule  in  any  state 
77  =  coefficient  of  viscosity 

xv 


xvi  NOTE 

C  =  coefficient  of  conduction  of  heat 
D  =  coefficient  of  diffusion 
6  =  rate  of  diffusion 

Zn  =  mean  momentum  transfer  distance 
Zc  =  mean  heat  transfer  distance 
/5  =  mean  diffusion  path 
$  =  interference  function 
K  =  path  factor 

d  =  projection  of  the  motion  of  a  molecule,  or  its  displace- 
ment 
t= period  of  displacement. 


A    KINETIC  THEORY 

OF 

GASES  AND  LIQUIDS 


CHAPTER    I 

THE    MOLECULAR    CONSTANTS,    AND    THE    DYNAMICAL 
PROPERTIES  OF  A   MOLECULE  IN  THE  GASEOUS  STATE 

1.  A  Brief  Historical  Summary  of  the  Develop- 
ment of  the  Kinetic  Theory  of  Explaining  the  External 
Pressure  and  other  Properties  of  Gases. 

In  order  to  explain  the  external  pressure,  or  elasticity  of  a 
gas,  vapor,  or  liquid,  it  is  necessary  to  introduce  a  theory, 
since  the  mechanism  giving  rise  to  this  property  is  not  evident 
to  the  eye.  A  purely  mechanical  explanation  based  on 
the  observed  elasticity  of  solid  materials,  or  of  a  mass  of 
fibrous  matter,  no  doubt  presented  itself  to  the  minds  of  the 
early  physicists.  But  since  from  the  earliest  times  some 
philosophers  regarded  matter  as  consisting  of  a  number  of 
hard,  indivisible,  and  similar  parts,  it  was  natural  that  it 
should  occur  to  some  to  explain  the  elasticity  of  a  gas  by  the 
motion  and  consequent  change  of  momentum  of  these  parts. 
Thus  Gassendi  in  the  17th  century  elaborated  an  atomic 
theory  of  the  properties  of  matter  based  upon  the  assump- 


2 


.  MOLECULAR  CONSTANTS 


tion  that  all  the  material  phenomena  can  be  referred  to  the 
indestructible  motion  of  atoms.  He  supposed  that  all  atoms 
are  the  same  in  substance,  but  different  in  size  and  form,  and 
that  they  move  in  all  directions  through  space.  A  number 
of  processes,  in  particular  the  transition  of  matter  from  one 
state  to  the  other,  were  explained  on  this  basis.  Later  these 
ideas  occurred  independently  to  other  investigators,  who 
elaborated  them  to  a  greater  extent.  Thus  Daniel  Bernoulli 
in  his  Hydrodynamica,  published  in  1738,  pointed  out  that 
the  elasticity  of  a  gas  may  be  explained  by  the  impact  of  the 
particles  of  which  it  was  supposed  to  consist  on  the  walls  of 
the  containing  vessel,  and  accordingly  he  deduced  Boyle's 
law  for  the  relation  between  pressure  and  volume.  Later 
the  subject  was  taken  up  by  Herapath,  Watertson,  Joule, 
Kronig,  and  with  great  success  by  Clausius.  Some  time 
later  Maxwell  added  some  important  contributions  to  the 
Kinetic  Theory.  In  the  hands  of  Clausius  and  Maxwell  it 
developed  with  great  rapidity  and  success.  The  subject 
now  attracted  numbers  of  theoretical  and  experimental  in- 
vestigators who  helped  to  perfect  it  theoretically,  and  by 
testing  the  results  experimentally  demonstrated  the  sound- 
ness of  the  underlying  assumptions.  Of  the  theoretical 
investigators  Boltzmann  should  be  specially  mentioned. 

The  endeavor  by  scientists  in  recent  years  to  extend  the 
well-known  mathematical  investigations  of  the  Kinetic 
Theory  of  Gases  to  liquids  and  dense  gases,  supposing  that 
molecular  interaction  may  be  represented  by  the  collision 
of  elastic  spheres,  has  been  almost  barren  of  results.  The 
object  of  this  book,  as  has  already  been  pointed  out  in  the 
Preface,  is  to  give  the  Theory  such  a  form  that  these  diffi- 
culties are  removed,  and  that  it  accordingly  applies  as  con- 
veniently to  liquids  as  to  gases.  Van  der  Waals  has  already 
rendered  important  service  by  means  of  his  theory  of  con- 
tinuity of  state,  in  indicating  the  general  nature  of  the  relation 
between  the  pressure,  volume,  and  temperature  of  a  sub- 


THE  ATOMISTIC  NATURE  OF  a  RAYS       3 

stance  in  any  state,  and  of  two  states  in  equilibrium  with  each 
other. 

In  this  development  a  knowledge  of  the  nature  of  the 
relative  distribution  of  matter  in  space  is  of  foremost  impor- 
tance, and  therefore  claims  first  attention. 

2.  A  Direct  Experimental  Proof  that  Matter 
consists  of  a  large  Number  of  Entities  or  Atoms. 

This  was  first  furnished  definitely  by  an  experiment 
devised  by  Rutherford  and  Geiger.*  It  was  arranged  that  a 
particles  of  radium  were  fired  through  a  gas  at  low  pressure, 
exposed  to  an  electric  field.  This  gave  rise  to  iomzation  in 
the  gas  which  was  measured  in  the  usual  way.  The  amount 
of  ionization  obtained  was  considerably  increased  by  in- 
creasing the  field  to  near  sparking  value.  The  velocity 
given  to  the  initial  ions  was  then  so  large  that  they  pro- 
duced further  ions  by  collision  with  neutral  molecules.  In 
this  way  the  small  ionization  produced  by  one  a  particle  in 
passing  through  the  gas  could  be  magnified  several  thousand 
times.  The  sudden  current  through  the  gas,  due  to  the 
entrance  of  a  single  a  particle  in  the  detecting  vessel,  was  by 
this  method  increased  sufficiently  to  give  an  easily  measur- 
able deflection  to  the  needle  of  an  ordinary  electrometer. 
Thus  by  limiting  the  number  of  a  particles  shot  into  the 
vessel  by  means  of  a  stop,  a  succession  of  throws  of  the  gal- 

*  Proc.  Roy.  Soc.  A.  81,  p.  141  (1908);  Phys.  Zeit.  10,  p.  1  (1909). 
NOTE.  There  existed  previously  a  good  deal  of  indirect  evidence 
that  matter  consists  of  entities.  Thus  the  law  of  constant  proportion, 
and  others,  in  chemistry  could  very  simply  be  explained  if  this  were 
the  case.  Also  the  formulae  obtained  for  the  viscosity,  conduction  of 
heat,  of  a  gas,  deduced  from  the  assumption  that  it  consists  of  particles 
of  matter  in  motion,  gave  a  general  agreement  with  the  facts.  But 
each  of  these  results  might  also  hold  on  mathematical  grounds  without 
matter  necessarily  consisting  of  atoms  or  molecules.  A  definite  proof 
of  the  atomistic  nature  of  matter  was  therefore  desirable  and  of  impor- 
tance, and  this  was  first  furnished  by  the  experiments  quoted. 


4  THE  MOLECULAR  CONSTANTS 

vanometer  needle  was  obtained.  This  proved  the  atomistic 
nature  of  the  a  radiation  given  off  by  radium. 

Ramsay  and  Soddy*  had  previously  shown  that  helium 
was  produced  from  radium  emanation,  and  that  it  is  there- 
fore one  of  the  products  of  the  disintegration  of  radium. 
It  was  suspected  that  helium  might  consist  of  a  particles 
which  have  lost  their  electric  charge.  This  was  proved  by 
Rutherford  and  Roydsf  who  showed  that  accumulated  a 
particles,  quite  independently  of  the  matter  from  which 
they  were  expelled,  consist  of  helium.  The  radioactive 
material  was  enclosed  in  a  glass  tube  whose  walls  were  so 
thin  that  they  were  penetrated  by  the  expelled  a  particles, 
which  were  retained  by  a  vessel  containing  the  glass  tube. 
The  matter  collected  in  this  way  was  tested  spectroscopically 
and  otherwise,  and  found  to  consist  of  the  gas  helium.  The 
atomistic  nature  of  one  of  the  gases  was  thus  proved;  and 
since  different  gases  possess  similar  physical  properties  this 
result  is  likely  to  hold  for  all  of  them. 

Some  interesting  experiments  by  DunoyerJ  may  be  men- 
tioned which  can  only  be.  reasonably  explained  if  matter 
consists  of  entities  which  possess  motion  of  translation. 
A  cylindrical  tube  was  divided  into  three  compartments 
by  means  of  two  partitions  perpendicular  to  the  axis  of  the 
tube,  each  partition  being  pierced  centrally  by  a  small  hole 
so  as  to  form  a  diaphragm.  The  tube  was  fixed  with  the  axis 
vertical  and  a  piece  of  some  substance  such  as  sodium, 
which  is  solid  at  ordinary  temperatures,  placed  at  the  bottom 
of  the  lowest  compartment.  The  tube  was  now  exhausted 
and  the  substance  heated  to  a  sufficient  temperature  to 
vaporize  it.  The  vaporized  particles  were  projected  in  all 

*  Nature,  68,  p.  246  (1903);  Proc.  Roy.  Soc.,  A.  72,  p.  204  (1903); 
73,  p.  346  (1904). 

t  Phil.  Mag.,  17,  p.  281  (1909). 

tComptes  Rendus,  152  (1911),  p.  592;  Le  Radium,Vlll  (1911),  p. 
142. 


THE  CHARGE  ON  THE  ELECTRON  5 

directions,  some  of  which  passed  through  the  hole  of  the  first 
diaphragm  at  various  angles.  Of  these  some  passed  through 
the  hole  in  the  second  diaphragm  forming  a  deposit  on  the 
top  of  the  tube.  This  deposit  was  found  to  coincide  exactly 
with  the  projection  of  the  hole  in  the  second  diaphragm 
formed  by  radii  drawn  from  the  hole  in  the  first  diaphragm. 
A  small  obstacle  placed  in  the  path  of  the  particles  was  found 
to  form  a  shadow  on  the  upper  surface  of  the  tube.  The 
vaporized  matter  thus  moved  along  straight  lines,  and  there- 
fore could  conceivably  consist -only  of  particles,  i.e.,  atoms. 

3.  The  Absolute  Mass  of  an  Atom  is  Most  Ac- 
curately Determined  from  a  Knowledge  of  the  Electric 
Charge  e  Carried  by  an  Electron. 

For  this  reason,  and  that  the  accurate  determination  of 
the  value  of  e  is  important  in  itself,  a  number  of  different 
principles  and  methods  have  been  employed  in  recent  years 
to  determine  this  quantity  with  the  greatest  possible  ac- 
curacy. 

A  determination  of  e  from  radioactive  data  was  made  by 
Rutherford  and  Geiger.*  The  number  of  a  particles  passing 
into  a  vessel  of  the  kind  mentioned  in  the  previous  Section 
were  directly  counted,  and  also  the  total  electrical  charge 
carried  by  the  particles  determined.  It  was  thus  found  that 
each  particle  carried  a  charge  9.3  X  10~10  units,  and  from 
various  evidence  it  was  concluded  that  this  was  twice  the 
unit  charge.  Similar  observations  were  carried  out  by 
Regener,f  who  counted  the  particles  by  noting  the  scintilla- 
tions that  they  produced  on  impinging  on  a  diamond.  The 
value  found  by  him  was  <?  =  4.79X10~10. 

MillikanJ  perfected  a  method  in  which  minute  drops  of 

*  Proc.  Roy.  Soc.,  A  81,  p.  162  (1908). 

^Ber.  d.  K.  Preuss.  Akad.  d.  Wiss.,  38,  p.  948  (1909). 

%  Phil.  Mag.,  19,  p.  209  (1910);  Science,  32,  p.  436  (1910). 


6          THE  MOLECULAR  CONSTANTS 

a  non-volatile  liquid  are  introduced  between  two  parallel 
and  horizontal  plates  having  a  strong  electric  field  between 
them.  The  drops  were  obtained  by  spraying  and  became 
electrically  charged  during  the  process.  The  movements 
of  individual  drops  under  the  action  of  gravity  and  the 
electric  field  were  observed  by  means  of  a  microscope.  By 
adjusting  the  electric  field  to  counteract  gravity,  it  was 
found  possible  to  keep  a  charged  drop  suspended  in  nearly 
the  same  position  in  the  gas  for  an  hour  at  a  time.  In  that 
case  if  X  denotes  the  electric  field,  N'e,  the  charge  on  a  drop 


where  r  denotes  the  radius  of  the  drop,  g  the  gravitational 
constant,  and  p  the  density  of  the  material.  The  value  of 
r  was  obtained  from  observations  of  the  rate  of  fall  of  the 
drop  under  the  action  of  gravity.  According  to  Stokes' 
equation  the  rate  of  fall  Vs  is  given  by 


where  rj  denotes  the  coefficient  of  viscosity  of  the  gas  through 
which  the  drop  moves.  A  number  of  experiments  were  made 
to  test  the  validity  of  Stokes'  equation  for  drops  of  different 
sizes,  and  a  suitable  correction  was  made  from  the  observ- 
ations. The  value  of  N'e  accordingly  could  be  determined. 
The  experiments  showed  that  each  of  these  drops  carried 
a  definite  multiple  of  a  small  charge,  which  accordingly 
is  equal  to  e.  The  final  result  obtained  gave  a  value*  e  =  4.77 
X10-10E.  S.  U. 

If  e  be  the  charge  of  electricity  carried  by  the  hydrogen 
atom  in  electrolysis,  and  N  the  number  of  atoms  in  one 

*  The  Electron,  by  R.  A.  Millikan.     The  University  of  Chicago 
Science  Series. 


THE  ABSOLUTE  MASS  OF  AN  ATOM  7 

gram  molecule  of  hydrogen,  it  is  known  from  experiments 
on  the  electrolysis  of  solutions  that 


Now  according  to  Rutherford  and  Geiger 
e  =  4.65X10-10E.S.U., 
or  e=1.55XlO-20E.M.U., 

and  accordingly 


which  is  also  the  number  of  molecules  in  a  gram  molecule 
of  any  substance,  according  to  the  definition  of  a  gram 
molecule.  The  absolute  mass  man  of  a  hydrogen  atom  is 
equal  to  1/N,  and  hence 

matf=1.60XlO-24grm. 

The   number   of    molecules   in  a  cubic  cm.  of  any  gas  at 
standard   pressure    and    temperature   can    now   be    shown 
simply  from  density  considerations  to  be  equal  to  2.78  X  1019. 
Millikan's  value  of  e  gives 

2V  =  6.06X1023, 
and 

=  1.65  X10-24  grin. 


The  remarkable  result  that  the  number  of  molecules 
in  a  gas  under  standard  conditions  is  a  contant  independent 
of  the  nature  of  the  gas,  and  the  fact  that  a  gas  as  a  whole  is 
compressible,  suggests  that  the  molecules  are  not  packed 
closely  together,  but  separated  from  each  other  by  distances 
much  greater  than  their  own  dimensions,  and  that  they 
undergo  niotion  of  translation  so  as  to  produce  on  the  aver- 
age an  even  distribution  of  matter  in  space,  and  give  rise 
to  the  external  pressure  of  the  gas, 


8  THE  MOLECULAR  CONSTANTS 

4.  Indirect  Experimental  Evidence  that  the  Mole- 
cules of  Gases  and  Liquids  are  in  Rapid  Motion. 

It  will  first  be  shown  that  there  are  definite  theoretical 
reasons  that  the  molecules  should  be  in  motion. 

Two  ways  only  suggest  themselves  of  explaining  the 
pressure  exerted  by  a  gas,  viz.:  (a)  By  forces  of  repulsion 
between  the  molecules,  (6)  by  a  molecular  motion  of  trans- 
lation. It  can  be  easily  shown  that  the  pressure  cannot  be 
exerted  according  to  (a) .  The  Joule-Thomson  effect  described 
in  Section  14  indicates  that  forces  of  attraction  exist  between 
molecules  which  act  over  distances  of  the  order  of  the  dis- 
tances of  separation  of  the  molecules  of  a  gas.  The  positive 
sign  of  the  internal  heat  of  evaporation  of  a  liquid  indicates 
that  this  also  holds  for  much  smaller  distances  of  separation 
of  the  molecules.  The  pressure  exerted  by  a  gas  cannot 
therefore  be  produced  by  forces  of  repulsion;  and  the 
molecules  must  therefore  possess  motion  of  translation  to 
account  for  the  pressure. 

It  will  be  shown  in  Section  6  from  dynamical  considera- 
tions that  the  magnitude  of  this  motion  decreases  as  the 
mass  of  the  molecule  increases.  Accordingly  if  the  mole- 
cules were  sufficiently  large  to  be  visible  to  the  eye,  we 
might  expect  that  this  motion  could  actually  be  observed. 
It  is  evident  that  the  motion  will  not  be  along  a  continuous 
straight  line,  but  zig-zag  shaped,  since  the  molecules  will 
collide  with  each  other.  This  will  also  be  the  case  with  a 
particle,  or  a  conglomeration  of  thousands  of  atoms,  con- 
tained in  a  liquid  or  gas,  for  the  particle  would  be  struck  on 
account  of  its  size  by  a  large  number  of  molecules  at  the 
same  instant,  but  the  same  number  would  not  necessarily 
strike  it  on  each  of  two  opposite  sides,  and  thus  the  motion 
of  the  particle  would  be  rendered  undulatory  in  character. 
These  conclusions  are  borne  out  in  a  striking  way  by  experi- 
ment. 


BROWNIAN  MOTION  9 

On  observing  by  means  of  a  microscope  small  particles 
of  matter  suspended  in  a  liquid,  they  are  found  to  be  under- 
going rapid  oscillatory  motion.  This  was  first  studied  by 
Brown  in  1827  in  the  case  of  pollen  of  plants  suspended  in 
water,  and  hence  has  been  called  Brownian  motion.  The 
motion  is  best  observed  by  means  of  the  ultra-microscope 
introduced  by  Siedentopf  and  Zsigmondy.*  In  this  appa- 
ratus a  parallel  beam  of  sunlight,  or  arc-light,  is  passed 
through  the  liquid  under  investigation  at  right  angles  to  the 
axis  of  the  microscope.  If  the  liquid  is  completely  homo- 
geneous no  scattered  light  enters  the  microscope.  On  the 
other  hand  if  the  liquid  contains  particles  of  matter  these 
appear,  through  scattering  some  of  the  light  passing  through 
the  liquid,  as  bright  specks  of  light  against  a  dark  back- 
ground. These  specks  of  light  undergo  a  rapid  oscillatory 
motion  indicating  the  motion  of  the  particles.  Through 
the  contrast  of  the  light  and  darkness  a  particle  becomes 
much  more  conspicuous  in  this  arrangement  than  under 
ordinary  conditions,  though  a  "proper  view"  of  the  particle 
is  entirely  lost.  It  is  necessary  that  the  incident  light  be  as 
intense  as  possible,  since  the  amount  of  scattered  light 
increases  in  proportion.  Also  the  layer  of  liquid  should  be 
as  thin  as  possible  in  order  to  decrease  the  absorption  of  the 
scattered  light,  and  minimize  the  overlapping  of  the  scat- 
tered light  from  different  particles. 

The  amplitude  of  the  oscillation  of  a  particle  was  found 
to  depend  upon  its  size.  For  a  diameter  of  l^u,  or  .001  mm., 
of  the  particle,  the  amplitude  is  about  equal  to  1/z,  while 
for  a  diameter  of  3^  the  amplitude  is  practically  zero.  Par- 
ticles 10-40/uM  in  diameter  have  amplitudes  up  to  20,u. 
It  appears  from  a  discussion  f  of  the  ultra-microscope  that 
it  is  still  possible  to  detect  by  this  method  a  linear  magni- 
tude of  6X10~7  cm.  =  6  juju,  or  about  1/100  of  a  light  wave. 

*  Drud.  Ann.,  10,  p.  1  (1903). 
ftoco.  tit.,  p.  14  (1903). 


10  THE  MOLECULAR  CONSTANTS 

Brownian  motion  has  also  been  observed  in  the  case  of 
smoke  particles  in  air. 

Other  designs  of  the  ultra-microscope  have  been  intro- 
duced besides  the  foregoing,  of  which  that  by  Cotton  and 
Monton*  deserves  special  mention.  In  this  arrangement 
the  beam  of  light  is  passed  through  the  liquid  in  such  a 
manner  that  it  is  totally  reflected  from  a  cover  glass  placed 
on  top  of  the  liquid. 

Liquids  containing  small  particles  in  suspension  may  be 
prepared  in  various  ways:  by  means  of  chemical  reactions 
involving  precipitation,  etc.;  by  sparking  different  metal 
electrodes  in  a  liquid,  etc.  < 

Some  well-known  experiments  by  Crookes  also  show  that 
motion  is  associated  with  matter,  and  that  an  increase  in 
temperature  gives  rise  to  an  increase  in  this  motion.  Thus 
if  a  thin  vane  of  light  material  is  suspended  vertically  in  a 
tube  partly  exhausted  and  heat  radiation  is  allowed  to 
fall  upon  it  at  right  angles,  it  is  deflected  in  the  direction  of 
propagation  of  heat.  This  effect  is  strikingly  shown  by  an 
apparatus  consisting  of  a  number  of  vanes  mounted  on  a 
horizontal  frame  which  can  rotate  round  a  vertical  axis  in 
a  partly  exhausted  tube,  the  planes  of  the  vanes  pass  through 
the  axis  of  rotation  and  each  alternate  side  is  covered 
with  lampblack  so  as  to  make  one  side  of  each  vane  a  better 
absorber  of  heat  than  the  other  side.  This  apparatus  is 
known  as  Crookes'  radiometer.  An  observer  in  the  same 
horizontal  plane  as  the  frame  carrying  the  vanes  is  therefore 
faced  by  blackened  vanes  on  one  side  of  the  axis  of  rotation 
and  by  unblackened  vanes  on  the  other  side  of  the  axis. 
If  heat  radiation  from  a  source  in  this  plane  is  allowed  to 
fall  on  the  arrangement  it  rotates  rapidly  in  a  direction  as  if 
a  greater  pressure  were  applied  to  the  blackened  surfaces  than 
to  the  unblackened  ones.  This  is  caused  by  the  temperature 
of  the  vanes  being  raised  above  that  of  the  gas  through  the 
*  Comptes  Rendus,  136,  p.  1657  (1903). 


DYNAMICAL  EFFECTS  OF  MOLECULES      11 

absorption  of  heat  radiation,  and  hence  heat  being  con- 
veyed from  the  vanes  to  the  gas  by  conduction.  Therefore 
if  heat  consists  of  kinetic  energy,  motion  is  given  to  the  gas 
in  the  immediate  neighborhood  of  the  vanes,  which  by 
reaction  gives  rise  to  pressures  acting  upon  them.  Now 
the  absorption  of  heat  by  the  blackened  sides  of  the  vanes 
is  greater  than  by  the  unblackened  sides,  and  hence  the  rise 
in  temperature  is  greater  in  the  former  case,  giving  rise  to  a 
greater  transfer  of  kinetic  energy  to  the  gas,  which  results 
in  a  greater  pressure  on  the  blackened  sides  than  on  the 
unblackened  ones.  Therefore  since  the  surface  of  a  vane 
rapidly  loses  the  absorbed  heat  when  not  exposed  to  the 
source  a  continual  motion  of  the  system  results. 

Further  information  as  a  guide  to  the  development  of  a 
kinetic  theory  of  matter  is  obtained  from  a  consideration 
of  the  physical  properties  of  a  gas  as  a  whole,  which  will  be 
described  in  the  next  Section. 

5.  The  Absolute  Temperature;  and  the  Equation 
of  a  Perfect  Gas,  and  of  a  Mixture  of  Gases. 

Experiments  on  the  coefficient  of  expansion  of  a  gas  at 
constant  pressure  have  shown  that  it  is  practically  inde- 
pendent of  the  pressure,  the  temperature,  and  the  nature 
of  the  gas,  and  is  approximately  equal  to  .0036625.  There- 
fore if  va  and  v  denote  the  initial  and  final  volume  of  a  gas 
when  its  temperature  is  changed  by  t°,  we  have 

......     (2) 


If  the  volume  of  the  gas  is  kept  constant  and  the  coefficient 
of  increase  of  pressure  is  measured,  it  is  found  to  be  inde- 
pendent of  the  volume,  temperature,  and  nature  of  the  gas, 
and  practically  the  same  as  the  foregoing  coefficient,  and 
thus 

p*po(H-.0036625t°),  .....  (3) 


12  THE  MOLECULAR  CONSTANTS 

where  p  and  po  denote  the  final  and  initial  pressures  respec- 
tively when  the  temperature  is  changed  by  t°. 

If  we  consider  a  mass  of  gas  whose  pressure  is  po  at  the 
temperature  0°  C.,  and  imagine  the  volume  kept  constant 
while  the  temperature  is  lowered  to  —  1°,  the  pressure  p 
will  be  given  by 


If  the  cooling  is  continued  to  a  temperature  of  —  1/.0036625 
or  -273°  C.,  then 

p=po(l-l)=0, 

i.e.  the  gas  would  exert  no  pressure  on  the  walls  of  the  con- 
taining vessel  at  this  temperature.  According  to  the  previous 
Section  this  can  only  occur  when  the  velocity  of  translation  of 
the  molecules  is  zero.  The  temperature  —273°  C.  has  accord- 
ingly been  called  the  absolute  zero.  The  temperature  of  a 
substance  may  be  measured  from  this  zero  and  denoted  by 
T,  in  which  case  0°  C.  corresponds  to  273  °C.,  and  in  general 
T°  =  273  +t°.  The  foregoing  equations  then  give 


and 

pccT, 

while  the  experiments  on  the  relation  between  p  and  v  at 
constant  temperature  give 

•"? 

which  is  Boyle's  law.    Accordingly 

pvccT, 
or 

pv  =  RiT, 

where  R\  is  a  constant.    If  v  refers  to  a  mass  of  M  grams  of 
gas  we  may  write  R\  =  RzM]    experiment  then  shows  that 


THE  EQUATION  OF  A  PERFECT  GAS  13 

R2  for  different  gases  varies  inversely  as  m  the  molecular 
weight  in  terms  of  that  of  the  hydrogen  atom.  Hence 
finally  the  equation  of  a  gas  becomes 

MET 


where  R  is  an  absolute  constant,  and  v  refers  to  the  volume 
of  a  gas  of  mass  M.  If  p  denote  the  density  of  the  gas  it 
follows  that  vp  =  M.  The  foregoing  equation  may  also  be 
written 

p  RT     ,r  RT 

P=—-—=NC-W,    .....   (5) 

ma  m  Ar 


where  N  denotes  the  number  of  molecules  in  a  gram  mole- 
cule, Nc  the  molecular  concentration,  and  ma  the  absolute 
molecular  weight  of  a  molecule. 

In  the  case  of  a  mixture  of  molecules  e  and  r  the  total 
pressure  p  according  to  equation  (5)  is  given  by 

•\r  RT  .  ,T  Rl      -*T  R  L 


where  pe  and  pr  denote  the  partial  pressures,  Ne  and  Nr  the 
partial  concentrations,  and  Ner  the  total  concentration  of 
the  molecules.  This  equation  is  the  same  in  form  as  equa- 
tion (5),  which  therefore  also  holds  for  a  mixture  of  any 
number  of  constituents. 

The  value  of  R  may  be  determined  by  means  of  the 
foregoing  equation  on  substituting  the  values  of  p,  v,  and  T, 
of  a  given  gas.  Since,  however,  in  practice  a  gas  does  not 
obey  strictly  equations  (2)  and  (3),  the  value  of  the  absolute 
zero  of  temperature  deduced  from  them  in  terms  of  any 
arbitrary  scale  of  temperature  requires  correction.  The 
thermodynamical  equation  (47)  which  involves  the  absolute 


14  THE  MOLECULAR  CONSTANTS 

temperature  is  usually  used  for  this  purpose.  If  t  denote 
the  temperature  readings  of  any  arbitrary  thermometer 
the  equation  may  be  written 

which  may  be  integrated  in  the  form 

fa 


T_  r_y 
T«  L  («L 


where  the  temperatures  T  and  To  on  the  absolute  scale 
correspond  to  t  and  to  on  the  arbitrary  scale.  The  quan- 
tities on  the  right  hand  side  of  this  equation  can  be  directly 

measured.     The  quantity   ( — )   is  most  conveniently  de- 

\  6v/t 

termined  from  experiments  on  the  Joule-Thomson  effect.* 
A  number  of  such  determinations  of  To  the  absolute  zero  have 
been  carried  out  by  a  number  of  observers;  in  the  Recueil 
de  Constantes  Physiques,  published  under  the  auspices  of  the 
Societe  Franchise  de  Physique,  the  most  probable  value  is 
taken  to  be 

To  =-  273.09  °C. 

From  values  of  p  and  v  for  oxygen  Berthelot  (Trav. 
et  Mem.  Bur.  Intl.)  obtains 

#  =  8.315X107. 

Equation  (4)  is  useful  in  helping  to  interpret  the  equa- 
tion for  the  velocity  of  translation  of  a  molecule  deduced 
from  purely  dynamical  considerations  in  the  next  Section. 

*For  further  information  see   Planck's    Thermodynamics,  pp.  12 
131,  and  Partington's  Thermodynamics,  pp,  162-167, 


THE  MOLECULAR  MOMENTUM  AND  GAS  PRESSURE     15 

6.  The  Velocity  of  Translation  of  a  Molecule  in 
a  Gas  from  Dynamics. 

Consider  a  single  molecule  of  absolute  mass  ma  moving 
to  and  fro  between  two  parallel  walls  at  right  angles  to  each 
with  a  velocity  V.  The  molecule  on  colliding  with  one  of  the 
walls  has  its  momentum  changed  by  2Vma,  and  since  this 
occurs  V/21  times  per  second,  where  I  denotes  the  distance 
between  the  walls,  the  change  in  momentum  per  second  is 

T7"O 

equal  to  —  =-A    Now  according  to  dynamics 

force  X  time  =  change  in  momentum, 

and  the  force  exerted  by  the  molecule  on  the  wall  is  there- 
fore equal  to  the  foregoing  expression.  Suppose  now  that 
there  are  Nc  molecules  per  cubic  cm.  moving  in  all  directions 
between  the  walls.  It  will  be  convenient  first  to  suppose 
that  these  molecules  consist  of  three  streams  each  equal 
to  Nc/3  moving  parallel  to  three  axes  at  right  angles  to  each 
other.  If  one  of  these  axes  is  taken  at  right  angles  to  the 
walls,  WVc/3  molecules  exert  the  foregoing  pressure  on  each 
square  cm.  of  the  walls.  Therefore  if  p  denotes  this  pres- 
sure we  have 

NcmaV2      V2 


where  Ncma  =  p,  the  density  of  the  gas. 

Let  us  now  suppose  that  the  molecules  move  in  all  direc- 
tions. Let  us  take  a  line  at  right  angles  to  the  walls  and  a 
point  on  the  line,  and  draw  a  sphere  of  radius  r  around  it. 
Molecules  will  pass  through  the  point  in  all  directions,  which 
is  equivalent  to  supposing  that  na  molecules  fall  per  square 
cm.  per  second  at  right  angles  on  the  surface  of  the  sphere. 
Let  us  find  an  expression  for  the  fraction  of  the  molecules 
passing  through  the  point  which  move  towards  one  of  the 


16          THE  MOLECULAR  CONSTANTS 

walls  and  make  an  angle  6  with  it.  On  taking  two  lines 
making  angles  6  and  6+dO  with  the  normal  and  rotating 
these  lines  with  the  normal  as  axis,  a  belt  whose  area  is 
2wr  sin  6-r-dQ  is  traced  out  on  the  sphere.  The  fraction  in 
question  is  therefore-  equal  to 

2irr  sin  6-r-dd-na       .     Q    -,Q 
-  =  sin  8  -  dd. 


Thus  of  the  total  number  of  molecules  Nt  between  the  walls 
Nt  sin  6  •  dd  molecules  make  an  angle  6  with  a  normal  to  one 
of  the  walls  and  move  towards  it.  Each  of  these  molecules 
on  striking  the  wall  gives  rise  to  a  change  in  momentum 

V  cos  6  ,  . 
equal  to  2maV  cos  6  and  this  happens  —  —  —  times  per  sec- 

ond through  rebounding  from  the  two  walls.  The  molecules 
under  consideration  therefore  exert  a  pressure  equal  to 

'  wne-.de 


I 

upon  the  wall.  The  pressure  P  exerted  on  the  whole  wall  by 
the  total  number  of  molecules  moving  in  all  directions  is 
therefore  given  by 


cos2  0sin  6-dd 


~m"  i  Jo' 

=  _m^    cos_ 


31 


If  Aa  denotes  the  area  of  the  wall,  p  the  pressure  per  square 
cm.,  and  Nc  the  number  of  molecules  per  cubic  cm.,  we  have 


By  means  of  these  two  equations  the  preceding  equation 
may  be  reduced  to  equation  (6).    It  appears  therefore  that 


THE  MOLECULAR  VELOCITY  OF  TRANSLATION      17 

we  may  represent  the  molecules,  as  far  as  the  pressure  they 
exert  is  concerned,  by  three  streams  moving  at  right  angles 
to  each  other. 

Since  p  and  p  can  be  directly  measured,  the  velocity 
of  translation  of  a  molecule  is  at  once  given  by 


(7) 


If  p  is  expressed  in  dynes  and  p  in  grams  per  cc.  the  velocity 
V  is  expressed  in  cms.  per  second. 

In  illustration  of  this  formula  the  values  of  V  in  cms. 
per  second  for  a  few  gases  at  standard  temperature  and 
pressure  are  given  by  Table  I. 


TABLE  I 


Substance. 

Formula. 

F9ms- 
sec. 

Hydrogen 

H2 

169  200 

Helium  
Methane    .                         

He 
CH4 

120,400 
60060 

Ethylene 

C2H4 

45  420 

Carbon  dioxide  

CO2 

36,250 

Amyl  propionate  

20,030 

C.  tetrachloride      .    .            .... 

CC14 

19390 

Ethyl  iodide 

C2H5I 

18820 

Mercury 

He 

17000 

Since  in  the  foregoing  investigation  the  velocity  of  a 
molecule  does  not  depend  on  the  presence  of  another  mole- 
cule, V  is  independent  of  the  volume  of  the  gas.  Therefore 
according  to  equation  (6)  the  pressure  of  a  gas  is  propor- 
tional to  its  density,  or  inversely  proportional  to  its  volume, 
which  agrees  with  the  facts  and  is  known  as  Boyle's  law. 


18 


THE  MOLECULAR  CONSTANTS 


By  means  of  equation  (4)  the  foregoing  equation  may 
be  written 


(8) 


The  velocity  of  translation  of  a  molecule  thus  varies 
inversely  as  the  square  root  of  its  mass,  and  hence  the 
greater  the  mass  the  smaller  the  velocity.  It  is  due  to  this 
that  it  is  possible  to  observe  directly  the  motion  of  large 
particles  in  a  liquid,  as  described  in  Section  4.  The  velocity 
is  also  proportional  to  the  square  root  of  the  absolute  tem- 
perature, and  hence  is  equal  to  zero  at  the  absolute  zero  of 
temperature. 


FIG.  1. 

In  the  foregoing  investigation  it  has  been  tac^Jy  assumed 
that  the  apparent  velocity  of  translation  of  a  molecule  is 
not  effected  by  the  presence  of  other  molecules.  This,  how- 
ever, does  not  hold;  the  velocity  is  apparently  increased  by 
the  volumes  of  the  molecules.  Thus  consider  two  molecules, 
a  and  6,  which  collide  head  on  so  that  each  retraces  its  path 
as  shown  in  Fig.  1.  This  amounts  to  the  same  thing  as  if 
the  molecules  at  the  moment  of  collision  were  to  be  inter- 
changed, the  molecule  a  then  taking  the  place  of  the.  retreat- 
ing molecule  b,  and  the  molecule  b  the  place  of  the  retreat- 
ing molecule  a.  The  velocity  of  each  molecule  is  thus  ap- 
parently increased,  since  each  passes  instantaneously  ovei 


THE  MOLECULAR  VOLUME  AND  GAS  PRESSURE     19 

a  part  of  its  path  equal  to  the  diameter  of  a  molecule.  Under 
these  conditions  the  walls  are  struck  oftener  per  second  by 
each  molecule,  giving  rise  to  a  greater  pressure  than  would 
be  obtained  if  each  molecule  had  no  volume  or  space  asso- 
ciated with  it  through  which  another  molecule  cannot  pass. 
The  values  of  V  given  by  equation  (7)  are  therefore  greater 
than  they  should  be.  It  is  easy  to  see,  however,  that  if  the 
diameter  of  a  molecule  is  small  in  comparison  with  the 
distance  of  separation  of  the  molecules,  as  is  the  case  with 
a  gas  under  ordinary  conditions,  the  error  introduced  in 
this  way  is  small.  It  will  be  evident  now  that  the  deviation 
of  a  gas  from  equation  (7)  is  in  part  caused  by  the  apparent 
volume  associated  with  each  molecule.  The  remaining  part 
of  the  deviation  is  caused  by  the  molecular  forces,  which 
will  be  discussed  in  Sections  14,  15,  and  21. 

It  has  also  been  assumed  that  each  molecule  possesses  the 
same  velocity.  This  does  not  hold  in  practice  in  a  gas,  since 
the  interaction  between  the  molecules  tends  to  continually 
change  their  velocities.  A  theoretical  investigation  showing 
that  this  is  the  case,  and  the  law  of  distribution  of  velocities 
obtained  by  Maxwell,  will  be  given  in  the  next  Section. 
But  we  may  still  express  p  by  means  of  equation  (6),  where 
V  is  now  called  the  average  kinetic  energy  velocity,  a  quan- 
tity which  will  be  discussed  in  Section  8.  It  is  the  velocity 
which  gives  the  average  kinetic  energy  of  a  molecule. 

7.  Maxwell's  Law  of  Distribution  of  the  Veloci- 
ties of  the  Molecules  of  a  Gas. 

It  can  be  shown  theoretically  that  the  molecules  in  a 
gas  at  any  instant  have  not  the  same  velocity;  moreover, 
the  velocity  of  each  molecule  is  continually  changing  in 
magnitude.  This  is  what  we  would  expect  on  account  of 
the  interaction  between  the  molecules.  According  to  Max- 
well's law,  which  will  be  proved  presently,  the  number  of 


20  .        THE  MOLECULAR  CONSTANTS 

molecules  N\  of  N  molecules  which  have  a  velocity  lying 
between  V\  and  V\+dV\  at  any  instant  is  given  by 

4A/Ti2       (Vl\z 

Nl,*"g-e-(vj  •dV1  .....     (9) 
VwVp* 

The  most  probable  velocity  is  the  value  of  V\  which  gives 
the  foregoing  expression  a  maximum  value.  Therefore  if 
the  differential  coefficient  of  the  expression  with  respect  to 
FI  is  obtained  and  equated  to  zero  we  have 

Vi  =  V,    .......     (10) 


Thus  Vp  is  the  most  probable  velocity  that  occurs  amongst 
the  gas  molecules. 

The  probability  of  the  occurrence  of  other  velocities 
decreases  rapidly  on  increasing  or  decreasing  the  value  of 
FI  from  Vp.  That  a  molecule  may  have  an  infinitely  small 
velocity  is  of  zero  probability,  and  this  also  holds  for  an 
infinitely  large  velocity.  It  appears  that  by  far  the  larger 
number  of  molecules  have  velocities  differing  but  little 
from  the  most  probable  velocity,  and  as  a  first  approxima- 
tion we  may  therefore  suppose  that  the  molecules  have  the 
same  velocity. 

The  most  probable  velocity  Vp  must  not  be  confounded 
with  the  average  velocity  Va-  The  latter  quantity  is  obtained 
by  adding  up  all  possible  velocities  and  dividing  their  num- 
ber into  the  sum  obtained.  This  corresponds  to  multiply- 
ing the  expression  for  NI  from  equation  (9)  by  FI,  integrat- 
ing it  between  the  limits  Fi=0and  FI=  oo,  and  dividing 
by  N,  giving 


/*» 
( 

o 


4F  v\     x2e~**  ,     C 

ft  T   +Jxe 


THE  PERMANENCE  OF  OSMOTIC  PRESSURE         21 

on  writing  V\/Vp=x.  Thus  the  average  velocity  is  greater 
than  the  most  probable. K 

Maxwell's  law  can  at  once  be  strictly  deduced  if  it  can 
be  shown  that  the  velocity  components  of  a  molecule  along 
three  axes  at  right  angles  are  independent  of  each  other. 
It  will  be  recognized  that  this  need  not  necessarily  hold 
without  being  proved,  on  account  of  the  interaction  between 
the  molecules.  Maxwell's  law  has  therefore  been  deduced 
by  a  number  of  mathematicians  in  different  ways  from  a 
general  investigation  of  molecular  motion,  but  which  all 
involve  one  or  more  assumptions  which  are  stated,  or  partly 
hidden  in  the  deductions.  Mathematicians  are  therefore 
not  yet  agreed  that  a  perfect  proof  has  been  given,  while 
some  maintain  that  the  law  cannot  hold.  The  writer  has 
developed  a  thermodynamical  proof  that  the  velocity  com- 
ponents mentioned  should  be  independent,  and  on  this  the 
proof  of  Maxwell's  law  will  be  based  in  this  book. 

It  can  be  shown  from  thermodynamics  that  osmotic 
equilibrium  is  not  of  a  temporary  nature,  but  must  be  per- 
manent. A  number  of  thermodynamical  formulae  have  been 
deduced  involving  the  isothermal  separation  of  substances 
by  means  of  semipermeable  membranes.  Van't  HofFs  formula 
for  the  heat  of  formation  of  a  substance  in  the  gaseous 
state,  for  example,  depends  upon  the  properties  of  semi- 
permeable  membranes.  The  thermodynamical  equation 
(47)  in  Section  21  may  evidently  be  applied  to  a  system 
involving  semipermeable  membranes.  Now  the  isothermal 
separation  of  substances  must  take  place  infinitely  slowly 
only,  otherwise  the  process  would  not  be  isothermal.  There- 
fore the  gases  on  the  two  sides  of  the  semipermeable  mem- 
brane of  a  system  must  be  in  permanent  equilibrium  with 
each  other  when  the  movable  parts  of  the  system  are  kept 
in  a  fixed  position,  that  is,  in  the  limiting  case  when  the 
movable  parts  are  kept  in  fixed  positions  the  diffusion  of 
the  different  gaseous  constituents  from  one  side  of  the 


22          THE  MOLECULAR  CONSTANTS 

membrane  to  the  other  must  be  infinitely  small.  If  this  were 
not  the  case  the  formulae  deduced  would  not  apply  in  prac- 
tice. Thus  in  the  limiting  case  the  diffusion  of  certain  gases 
through  a  given  membrane  may  be  infinitely  small,  while 
other  gases  pass  readily.  Each  of  the  constituents  of  a 
gaseous  mixture  may  thus  exist  on  the  two  sides  of  a  mem- 
brane, when  initially  placed  on  one  side,  but  the  condition 
of  the  system  may  be  such  that  some  of  them  exist  on  the 
other  side  of  the  membrane  in  infinitely  small  amounts  only, 
which  condition  remains  unaltered. 

The  partial  pressure  of  each  constituent  of  a  mixture  of 
gases  is  the  same  as  if  the  other  gases  were  absent.  Hence 
the  redistribution  of  the  velocities  of  the  molecules  of  a 
gas  that  might  take  place  on  addition  of  another  gas  is  such 
that  each  molecule  produces  the  same  average  pressure  as 
before,  which  is  equivalent  to  each  molecule  having  the 
constant  velocity  V  given  by  equation  (7),  which  will  be 
called  the  pressure  velocity  in  this  Section. 

Let  us  now  consider  a  mixture  of  molecules  a  and  b  in 
a  chamber  A  B  separated  from  a  chamber  A  by  a  semi- 
permeable  membrane,  which  is  permeable  only  to  the 
molecules  a.  A  migration  of  the  molecules  a  from  the  cham- 
ber A  through  the  membrane  into  the  chamber  AB,  and  in 
the  reverse  direction  is  continually  going  on.  This  is  shown 
by  the  fact  that  on  increasing  the  volume  of  the  chamber 
A  molecules  a  at  once  pass  into  it  from  the  chamber  AB,  and 
that  therefore  there  is  a  free  passage  for  the  molecules  a 
through  the  membrane  which  can  only  be  interfered  with 
occasionally  by  molecules  coming  in  the  opposite  direction. 
The  number  of  molecules  passing  in  one  direction  per  second 
is  evidently  equal  to  the  number  of  molecules  passing  in  the 
opposite  direction,  otherwise  the  number  of  molecules 
on  each  side  of  the  membrane  would  gradually  change. 
We  have  seen  that  the  pressure  velocity  V  of  translation  of 
the  molecules  a  is  the  same  in  the  two  chambers.  The 


THE  INDEPENDENCE  OF  MOLECULAR  VELOCITIES     23 

pressure  velocity  of  each  set  of  migrating  molecules  must 
also  be  the  same  and  equal  to  the  foregoing  velocity  *  other- 
wise the  pressure  would  gradually  change  in  the  two  cham- 
bers. Let  N'  denote  the  number  of  molecules  migrating 
from  one  chamber  into  the  other  when  no  molecules  b  are 
in  the  chamber  AB,  and  N"  the  number  corresponding  to 
a  given  number  of  molecules  b  in  the  chamber.  Thus  the 
addition  of  the  molecules  b  to  the  chamber  AB  has  the  effect 
of  dividing  the  number  of  molecules  N'  into  two  parts,  the 
part  N"  which  migrates  and  the  part  N'— N"  which  does  not, 
the  pressure  molecular  velocity  being  the  same  for  each  part 
according  to  what  has  gone  before.  Now  a  set  of  molecules 
of  different  velocities  can  be  divided  in  one  way  only  into 
two  parts  satisfying  the  foregoing  conditions,  viz.:  which 
corresponds  to  the  same  distribution  of  velocities  in  each 
part.  For  if  the  number  of  molecules  N"  is  decreased  by 
n',  the  pressure  velocity  of  these  n'  molecules  must  be  equal 
to  V,  since  this  holds  for  the  molecules  N"  —  nf  and  N'— 
(N"  —  nf),  which  can  only  be  realized  in  the  foregoing  way 
unless  each  molecule  a  has  the  same  velocity.  \  Thus  the 
distribution  of  velocities  between  the  molecules  migrating 
from  the  chamber  A  into  the  chamber  AB  is  the  same  as 
that  of  the  molecules  in  the  chamber  A,  and  the  distribution 
amongst  the  molecules  migrating  from  the  chamber  AB 
into  the  chamber  A  is  the  same  as  that  of  the  molecules  in 
the  chamber  AB.  These  two  distributions  can  be  shown  to 
be  identical. »  If  this  were  not  the  case  the  molecules  passing 
into  the  chamber  A  would  give  rise  to  a  different  distribu- 
tion of  velocities  near  the  membrane  than  exists  at  other 
parts  of  the  chamber.  This  would  effect  the  migration  of 
the  molecules  a  from  the  chamber  A  into  the  chamber  A  B, 
and  thus  disturb  the  equilibrium.  *  From  this  it  follows 
that  the  distribution  of  velocities  between  the  molecules 
a  in  the  chamber  A  is  the  same  as  that  in  the  chamber  AB. 
The  addition  of  molecules  b  to  the  chamber  AB,  accord- 


24  THE  MOLECULAR  CONSTANTS 

ing  to  the  foregoing  investigation,  does  not  alter  the  dis- 
tribution of  the  velocities  between  the  molecules  a  from  that 
obtained  wjien  molecules  a  only  are  in  the  chamber.  Thus 
the  velocity  components  at  right  angles  to  each  other  of 
the  molecules  a  are  not  changed  by  collision  with  the  mole- 
cules 6,  and  hence  not  changed  by  the  collision  of  the  mole- 
cules a  with  each  other,  i.e.,  the  velocity  components  are 
independent  of  each  other.  This  result  will  now  be  used 
to  prove  Maxwell's  law. 

Let  the  rectangular  components  of  the  velocity  V\  of  a 
molecule  be  denoted  by  a,  6,  and  c,  in  which  case 


Let  the  probability  that  the  component  velocity  along  the 
x  axis  has  a  value  lying  between  a  and  a-\-da  be  expressed 
by  the  function  /(a),  similarly  let  /(&)  and  /(c)  express  the 
probabilities  that  the  component  velocities  along  the  y 
and  z  axes  lie  between  6  and  6+d6,  and  between  c  and  c+ 
dc,  respectively.  The  probability  of  the  three  components 
occurring  simultaneously  is  therefore  /(a)-/(6)-/(c),  since 
the  components  are  independent.  The  situation  of  the  sys- 
tem of  coordinates  in  space  is  arbitrary,  and  therefore 

/(a)  •/(&)  •/(«)  =  *i(«2+62+c2),     >/ 

where  <j>i  is  a  definite  function  of  Fi2.  On  differentiating 
this  equation,  keeping  Vi  constant,  and  dividing  the  result- 
ant equation  by  /(a)  •/(&)  -/(c),  we  obtain 


The  differentiation  of  equation  (12)  under  the  same  condi- 
tion gives 


THE  COMPONENT  VELOCITY  PROBABILITY          25 

On  multiplying  this  equation  by  the  undetermined  multi- 
plier X,  the  foregoing  two  equations  may  be  combined  into 
the  equation 


Since  the   changes  da,  db,  and  dc,   are   independent  their 
factors  may  be  separately  equated  to  zero,  giving 


and  two  similar  equations  involving  b  and  c.    The  integra- 

tion of  these  equations  gives  /(a)  =  C\e   ^°  ,  /(&)  =  C\e    2    , 

-Ac2 
and/(c)=Ci6    2   ,  where  Ci  denotes  an  arbitrary  constant 

whose  value  is  infinitely  small,  since  the  probability  of  a 
certain  case  in  an  infinitely  large  number  of  possible  cases 
is  infinitely  small.  We  may  therefore  write  Ci  =  C2-da, 

and  if  besides  we  write  ?>     \fT/    we  nave 

-(—  Y 
f(a)  =  C2e    \vpl  -da. 

Since  the  sum  of  the  probability  of  an  event  happening  and 
the  probability  of  it  failing  is  equal  to  unity,  we  have  for 
all  possible  cases 


( 

*/-oo 


C2  I     e    W  -da=l, 

the  integral  expressing  the  sum   of  all   the   probabilities. 
Similarly  we  have 


( 

J-co 


C2        e  •<#>=!, 


26  THE  MOLECULAR  CONSTANTS 

which  on  multiplying  by  the  preceding  equation  becomes 

/*«    r*    _oH-6_2 
C22  e     vp*  -da-db=l, 

J  -&J-<x> 

or 

C22VP2  (     (    e-(x*+^dx-dy=l, 

J  —  &J-V) 

if  we  write  a/Vp  =  x  and  b/Vp  =  y.  If  x  and  y  are  now 
regarded  as  coordinates  in  a  system  of  rectangular  coordinates 
they  may  be  transformed  into  polar  coordinates  by  writing 

x2+y2  =  r2  and  dx'dy  =  r'dr-d(}>, 
which  transforms  the  foregoing  integral  into 

C22V2 


2C     C"  e-r2 
Jo    Jo 


and  hence 

C2  =  - 

Accordingly 


1         (  b  Y 
-da,       /(6)  =  -*e-^-J  .db, 

VpVir 

1      -(-^-Y 


and  the  probability  that  the  three  components  a,  b,  and  c, 
occur  simultaneously  is  therefore 

1 


To  obtain  the  probability  of  the  occurrence  of  a  certain 
velocity  Vi  let  us  as  before  take  a,  6,  and  c,  as  coordinates, 
in  which  case 


THE  DISTRIBUTION  OF  MOLECULAR  VELOCITIES     27 

and 

da-db'dc=Vi2-dVi'  smO-dft-dcf), 

where  0  denotes  the  angle  between  V\  and  the  c  axis,  and 
0  the  angle  between  the  a  axis  and  the  projection  of  V\  on 
the  (a,  6)  plane.  The  probability  of  a  velocity  V\  having  a 
definite  direction  is  therefore 

1       -(^lY 

>/  FI  -dVi-  sin  Q-d8'd<t>. 


7T 


The  probability  independent  of  any  direction  is  obtained 
on  integrating  with  respect  to  6  from  0  to  TT,  and  with  respect 
to  $  from  0  to  27r,  which  is  taking  into  account  all  possible 
directions.  This  can  easily  be  shown  to  give 


which  expresses  the  probability  that  a  molecule  has  a  veloc- 
ity lying  between  Vi  and  V \-\-dV\.  The  fraction  of  N 
molecules  possessing  this  velocity  is  then  immediately  given 
by  equation  (9),  which  completes  the  proof. 

It  will  easily  be  seen  that  the  average  velocity  of  trans- 
lation of  a  molecule  depends  on  the  nature  of  the  distribu- 
tion of  the  velocities  amongst  the  molecules.  But  the  aver- 
age kinetic  energy  of  a  molecule  is  independent  of  this 
distribution,  as  will  appear  from  the  next  Section. 

8.  The  Average  Kinetic  Energy  Velocity  of  a 
Molecule,  and  the  Relation  between  the  Kinetic 
Energy  of  a  Molecule  and  its  Absolute  Temperature. 

If  Ni  molecules  in  a  cubic  cm.  of  a  gas  have  a  velocity 
Vit  and  N2  molecules  a  velocity  V2,  etc.,  where 

Ni+N2+  .      .  Nn=Nc, 


28  THE  MOLECULAR  CONSTANTS 

the  total  number  of  molecules  in  the  cubic  cm.,  the  average 
kinetic  energy  of  a  molecule  is  given  by 


,      (NiV1*+N2V22+  .  .  .  NnVn*\ 
*»•(-      Nl+N2+...Nn-      J' 

where  ma  denotes  the  absolute  mass  of  a  molecule.     This 
expression  for  the  kinetic  energy  may  also  be  written 


where  V  will  be  called  the  average  kinetic  energy  velocity 
of  a  molecule,  and  is  given  by 


AW-f 
\          tfi 


+N2V2*+  .  .  .  NnVn2 


.  .  .Nn 

This  velocity  is  evidently  not  equal  to  the  average  velocity 
Va,  which  is  given  by 

_N1V1+N2V2+  .  .  .NnVr 
Nl+N2+  .  .  .  Nn 

It  can  be  shown  that  the  velocity  V  is  independent  of  the 
distribution  of  the  molecular  velocities.  Thus  if  pi  denote 
the  partial  pressure  of  the  N\  molecules  having  a  velocity 
Vi,  and  p2  the  partial  pressure  of  the  N2  molecules  having 
a  velocity  V2,  etc.,  the  total  pressure  p  of  the  gas  is  given 
by 


But  according  to  equation  (6) 


MOLECULAR  ENERGY,  TEMPERATURE,  AND  MASS     29 
and 


pn=^-X2, 


and  hence 


This  equation,  which  is  same  as  equation  (6),  expresses  V 
in  terms  of  p  and  maj  and  it  is  therefore  independent  of  the 
distribution  of  molecular  velocities. 

The  kinetic  energy  of  a  gram  molecule  of  molecules  in 
the  gaseous  state  therefore  becomes 


=      T  =  yT=  1.247  Xl&T  ergs,    .     .     (13) 


by  the  help  of  equation  (8).  The  average  kinetic  energy 
of  a  single  molecule,  which  is  1/N,  or  1/6.2X1023,  the  fore- 
going value,  is  therefore  equal  to 


(14) 


The  values  of  R  and  N  used  are  given  in  Sections  5  and  3. 
Thus  the  average  kinetic  energy  of  a  molecule,  whatever  the 
distribution  of  molecular  velocities,  is  simply  proportional 
to  the  absolute  temperature,  and  thus  independent  of  molec- 
ular mass. 

It  will  be  of  interest  to  obtain  the  connection  between 
the  average  kinetic  energy  velocity  V  of  a  molecule  and  its 
most  probable  velocity  Vp  according  to  Maxwell's  law.  The 
number  of  molecules  of  Nc  molecules  in  a  cubic  cm.  whose 
velocities  lie  between  FI  and  V\+dV\  at  any  instant  is 
equal  to 

INcVi2     (IiV    ,v 
e-(Vp)-dV, 


30  THE  MOLECULAR  CONSTANTS 

according  to  the  preceding  Section.     The   pressure  these 
molecules  exert  is  obtained  by  multiplying  this  expression 

by  -~^l£i2,  according    to    equation  (6).     Therefore  on  inte- 

o 

grating  the  foregoing  product  from   Vi  =  Q  to  FI  =  QO  we 
obtain  the  total  pressure  p  exerted  by  the  molecules,  that  is 


IWaNcV*    C^  4 

—  —  -  I    x4e~x 

3V  7T       JQ 


~x  -dx 


3V7 


since         e   x -dx=-— ,  where  p  denotes  the  density.     On 

Jo  * 

comparing  the  foregoing  value  of  p  with  that  given  by  equa- 
tion (6)  we  obtain 


/Q\ 

=(l) 


or  the  average  kinetic  energy  velocity  is  about  23%  greater 
than  the  most  probable  velocity.  The  relation  between 
V  and  Va  is  obtained  from  a  comparison  of  the  foregoing 
equation  and  equation  (11),  which  gives 

8,, 


Thus  the   average  velocity  is   about   92%   of  the  average 
kinetic  energy  velocity. 

In  the  case  of  a  mixture  of  gases  of  molecules  r  and  e  the 
total  pressure  p  is  given  by 

e2   . 
1 


THE  EQUATION  OF  A  GASEOUS  MIXTURE  31 

on  applying  equation  (6)*,  where  pe  and  pr  denote  the  par- 
tial pressures,  Ne  and  Nr  the  partial  concentrations,  mae 
and  mar  the  absolute  molecular  weights,  and  Ve  and  Vr  the 
average  kinetic  energy  velocities  of  the  molecules  e  and  r 
respectively.  If  the  temperature  of  each  set  of  molecules 
is  the  same  an  application  of  equation  (8)  to  the  foregoing 
equation  gives 


where  Ner  denotes  the  total  concentration  of  the  molecules 
per  cubic  en.,  and  me/mae=nir/mar=N  the  number  of  mole- 
cules in  a  gram  molecule  of  a  pure  substance.  ,  This  equation, 
which  is  the  same  as  equation  (5),  agrees  with  the  facts, 
showing  that  when  two  gases  are  mixed  they  rapidly  assume 
the  same  temperature.  This  would  arise  in  part  through 
both  gases  being  in  contact  with  the  vessel  whose  tempera- 
ture they  would  gradually  assume,  in  other  words,  the 
vessel  usually  being  a  conductor  of  heat  would  act  as  an 
intermediary  in  adjusting  the  gases  to  the  same  temperature. 
The  question  then  arises,  would  an  equalization  of  tempera- 
ture take  place  through  the  interaction  of  the  molecules 
alone,  and  what  is  the  distribution  of  velocities  when  the 
two  sets  of  molecules  in  some  way  or  other  have  acquired 
the  same  temperature?  This  will  be  discussed  in  the  next 
Section. 

9.  The   Equipartition    of    Energy    between    the 
Different  Molecules  in  a  Mixture  of  Gases. 

When  two  sets  of  molecules  of  different  kinds  at  different 

temperatures  are  mixed,  the  average  kinetic  energy  of  each 

molecule  of  each  set,  according  to  the  Law  of  Equipartition 

of   Energy,    eventually  becomes   the   same  through  molec- 

*  We  may  do  this  since  each  set  of  molecules  acts  independently. 


32          THE  MOLECULAR  CONSTANTS 

ular  interaction,  and  the  corresponding  distribution  of  the 
molecular  velocities  at  any  instant  is  the  same  as  if  the  gases 
were  isolated.  This  law  is  usually  proved,  or  attempts  are 
made  to  prove  it,  from  purely  dynamical  considerations 
in  treatises  on  the  Kinetic  Theory  of  Gases.  The  analysis 
is  very  intricate,  and  the  suppositions  introduced  are  not 
unobjectionable.  The  subject  has  always  attracted  a  good 
deal  of  attention  from  mathematicians  with  the  object  of 
putting  the  proof  on  a  sounder  basis.  In  the  earlier  develop- 
ment of  the  subject  the  investigations  of  Maxwell  and 
Baltzmann  are  pre-eminent. 

The  dynamical  proof  of  the  law  usually  depends  on  the 
assumption  that  the  molecules  consist  of  perfectly  elastic 
spheres  not  surrounded  by  fields  of  force,  and  that  the 
equipartition  of  energy  is  solely  caused  by  molecular  collision. 
As  a  matter  of  fact  in  practice  the  equipartition  would  be 
caused  in  other  ways  if  not  caused  by  molecular  collision. 
Thus  we  know  from  experience  that  a  mass  of  a  hot  gas 
located  in  a  colder  gas  radiates  heat  to  the  latter  till  the 
temperature  is  equalized.  A  well-known  example  of  this  is 
the  radiation  manifested  by  the  hot  air  rising  from  a  fire, 
or  from  a  furnace.  This  shows  that  each  individual  mole- 
cule in  a  gas  is  continually  radiating  heat  energy  whose 
amount  per  second  depends  on  its  kinetic  energy.  There- 
fore if  two  sets  of  molecules  at  different  temperatures  are 
mixed  heat  will  be  radiated  from  one  set  of  molecules  to  the 
other  till  the  average  kinetic  energy  of  each  molecule  is  the 
same,  or  the  two  sets  of  molecules  have  the  same  tempera- 
ture, as  would  be  the  case  if  the  two  sets  of  molecules  were 
separate  but  adjacent  to  each  other.  It  will  follow  then  from 
Section  7  that  after  temperature  equilibrium  has  been 
obtained  the  distribution  of  velocities  in  each  set  of  mole- 
cules is  the  same  as  if  it  were  isolated. 

It  seems  unnecessary  and  futile,  therefore,  to  endeavor  to 
establish  the  Law  of  Equipartition  of  Energy  on  assumptions 


EQUIPARTITION  OF  ENERGY  33 

relating  to  the  interaction  of  molecules,  when  the  law  fol- 
lows directly  from  the  fact  that  a  molecule  is  continually 
radiating  heat  energy.  Although  equipartition  of  energy 
would  be  brought  about  as  indicated,  it  is  very  likely  that  it 
would  also  be  brought  about  by  the  collision  of  elastic  spheres, 
since  it  follows  from  mechanics  that  on  the  average  two 
colliding  bodies  of  different  kinetic  energies  have  their 
energies  more  evenly  distributed  after  collision.  In  fact, 
every  kind  of  interaction  between  two  molecules,  whether 
by  "  collision,"  or  through  the  molecular  forces  of  attrac- 
tion and  repulsion,  has  on  the  average  the  effect  of  redis- 
tributing the  kinetic  energy  in  favor  of  the  body  possessing 
the  lesser  energy.  A  definite  mathematical  proof  that 
equipartition  of  energy  may  take  place  along  these  lines  is 
difficult  on  account  of  having  to  deal  with  a  large  number 
of  molecules  possessing  different  velocities,  each  of  which 
interacts  some  time  with  one  or  more  molecules,  and  it  is 
therefore  necessary  to  show  that  equilibrium  in  energy 
partition  exists  after  an  infinite  time  beginning  from  an  in- 
definite state  of  affairs. 

10.  The  Number  of  Molecules  per  Cubic  Cm.  in 
a  Gas. 

According    to   equation   (5)   the  number  Nc  of   mole- 
cules per  cubic  cm.  of  a  gas  is  given  by 


where  N  and  R  are  given  in  Sections  3  and  5.  Hence  if 
two  different  gases  possess  the  same  temperature  and 
pressure  each  contains  the  same  number  of  molecules  per 
cubic  cm.  This  result  is  known  as  Avogadro's  Law.  It 
follows  also  that  two  gases  at  different  temperatures  con- 
tain the  same  number  of  molecules  per  cubic  cm.  if  the 
ratio  p/T  has  the  same  value  for  both.  It  will  appear 


34          THE  MOLECULAR  CONSTANTS 

from  Section  5  that  these  results  also  hold  for  mixtures  of 
gases,  in  which  case  p  denotes  the  sum  of  the  partial 
pressures. 

The  foregoing  equation  may  also  be  written 


,  (16) 

m  m 

according  to  equation  (5),  from  which  it  follows  that  two 
gases  not  necessarily  at  the  same  temperature  contain  the 
same  number  of  molecules  per  cubic  cm.  if  they  have  the 
same  values  for  the  ratio  p/m. 

Since  the  molecules  possess  motion  of  translation  this 
leads  us  to  the  consideration  of  another  quantity.  ^ 

11.  The  Number  of  Molecules  Crossing  a  Square 
Cm.  in  all  Directions  from  one  Side  to  the  Other  in  a 
Gas. 

This  number  is  of  importance,  since  it  is  one  of  the  funda- 
mental quantities  occurring  in  the  different  formulae  relating 
to  a  gas.  It  can  be  expressed  in  terms  of  quantities  which 
can  be  measured  directly,  and  hence  its  numerical  value 
obtained  when  desired.  We  will  see  in  Section  29  that  its 
value  can  also  be  found  in  the  case  of  a  liquid.  It  will  be 
convenient  first  to  obtain  the  number  corresponding  to  the 
supposition  that  the  molecules  move  parallel  to  three  axes 
at  right  angles  to  each  other.  We  have  seen  that  this  suppo- 
sition may  be  made  when  dealing  with  the  connection  between 
the  velocity  of  translation  of  the  molecules  and  the  pres- 
sure which  they  exert.  Let  no  denote  the  number  of  mole- 
cules crossing  a  plane  one  square  cm.  in  area  per  second  in 
one  direction  situated  at  right  angles  to  one  of  the  foregoing 
axes.  It  follows  then  that 


THE  PRESSURE  FACTORS  OF  A  MOLECULE          35 

where  Va  denotes  the  average  velocity  of  a  molecule,  Nc 
the  molecular  concentration,  and  p  the  density  of  the  gas. 
To  prove  this  we  may  suppose  that  we  are  dealing  with  a 
single  cubic  cm.  of  gas  whose  faces  are  parallel  to  the  axes 
mentioned,  in  which  case  a  molecule  having  a  velocity  Vx 
will  cross  it  Vx/2  times  in  one  direction  per  second.  There- 
fore if  there  are  Ni,  N2,  Ns  .  .  .  ,  molecules  having  respec- 
tively the  velocities  Vi,  ¥2,  V$  .  .  .  ,  we  have 


Ne(NlVi+NjV2+.      .}     NcV 


61  Nc  j         6 

Another  expression  for  no  whic,h  is  very  useful  may  be 
obtained.  The  pressure  p  in  dynes  exerted  by  a  gas  may  be 
written 

(18) 


where  A  represents  an  appropriate  factor,  which  has  other 
important  applications  which  will  be  found  in  Section  20. 
The  value  of  the  quantity  A  may  be  determined  by  substi- 
tuting in  the  foregoing  equation  successively  for  the  quan- 
tities p,  no  and  Va,  assuming  that  Va  =  V,  from  the  equations 
(17),  (8),  and  (4),  giving 

•     (19) 

where  wa/m=1.6lXlO~24,  the  absolute  value  of  the  mass 
of  a  hydrogen  atoin. 

Let  us  now  consider  the  case  when  the  molecules  are 
moving  in  all  directions.  Let  a  point  6  be  taken  on  a  plane 
abc  in  the  gas,  Fig.  2,  arid  a  sphere  of  radius  r  described 
round  it.  The  molecules  passing  through  the  point  6  strike 
the  surface  of  the  sphere  at  right  angles,  and  let  us  therefore 


36  THE  MOLECULAR  CONSTANTS 

suppose  that  their  number  corresponds  to  S  molecules 
entering  the  sphere  per  second  per  cm.2  of  its  surface.  The 
number  of  molecules  impinging  on  a  circular  belt  of  the 
sphere  of  breadth  r-dd,  and  radius  r  cos.  6  is  therefore  2irr2S 
cos  0  •  dO.  The  force  exerted  by  a  molecule  impinging  on  the 
plane  abc  at  an  angle  6  is  proportional  to  its  component 
velocity  at  right  angles  to  the  plane  according  to  equations 
(17)  and  (18),  and  thus  according  to  equation  (18)  equal  to 

— ^ — A,  or  sin  0-A.     Hence  the  molecules  entering  the 


abc 

FIG.  2. 

upper  hemisphere,  which  lie  in  the  solid  angle  TT,  on  impinging 
on  the  point  b  exert  the  force 

(  22Trr2SA  sin  0-  cos  6~de=Trr2SA. 
Jo 

If  u  denotes  this  number  of  molecules  we  have  u  =  2irr2S, 
and  the  pressure  exerted  may  therefore  be  written  u-A/2. 
Hence  if  n  denotes  the  number  of  molecules  crossing  a  square 
cm.  from  one  side  to  the  other  in  all  directions,  the  pressure  p 
of  the  gas  is  given  by 

.     .     ,'(20) 


CORRECTION  FACTORS  FROM  MAXWELL'S  LAW     37 

This  equation  may  be  used  to  find  n.    The  equation  may 
also  be  written  in  the  form 


(21) 


by  means  of  equations  (19)  and  (4),  where  v  denotes  the 
volume  of  a  gram  molecule  of  molecules. 

In  the  foregoing  investigation  we  have  assumed  that 
Va=V,  or  the  average  velocity  of  translation  of  a  molecule 
is  equal  to  the  average  kinetic  energy  velocity.  But  this  is 
not  the  case  according  to  Section  8.  According  to  Max- 

/~8~ 
well's  law  Va=+—V=.922V.    If  this  is  accepted  and  taken 

\O7T 

into  account  the  value  of  A  is  5.521  XlQ~20VTm,  or  1.085 
times  that  given  by  equation  (19).  In  subsequent  investi- 
gations we  shall,  however,  use  the  value  of  A  given  by 
equation  (19).  The  necessary  change  if  desired  can  always 
be  readily  made. 

The  value  of  n  given  by  equation  (21)  is  corrected  accord- 
ing to  Maxwell's  law  by  multiplying  the  right-hand  side 

/~8~ 
of  the  equation  by  */*-i  or  by  .922. 

Since  a  molecule  in  motion  represents  a  certain  amount 
of  kinetic  energy,  general  energy  considerations  will  be  of 
interest  and  importance. 

12.  The  First  Law  of  Thermodynamics. 

According  to  this  law  energy  is  indestructible.  Since  the 
expenditure  of  energy  on  a  substance  is  accompanied  by  a 
change  in  temperature  heat  represents  a  form  of  energy. 
The  amount  of  change  in  temperature  depends  on  the  heat 
capacity,  or  the  specific  heat,  of  the  substance.  By  defini- 
tion the  heat  capacity  of  water  between  the  temperatures 


38          THE  MOLECULAR  CONSTANTS 

14.5°  C.  and  15.5°  C.  is  unity,  and  the  amount  of  heat  ab- 
sorbed corresponding  to  this  change  in  temperature  is  called 
a  •calorie.  Other  definitions  of  the  calorie  have  been  pro- 
posed, but  the  foregoing  is  mostly  used.  The  units  of  heat 
capacity  and  mechanical  work  are  quite  arbitrary,  and  there- 
fore a  factor  according  to  the  above  law  should  exist  which 
would  enable  heat  units  to  be  converted  into  mechanical 
units,  and  vice  versa,  that  is,  if  W  denotes  the  amount  of 
work  in  ergs  expended  to  produce  an  amount  of  heat  Q  in 
calories,  the  relation  between  the  two  quantities  is  expressed 
by  the  equation 

W  =  JQ, 

where  J  denotes  the  factor  in  question. 

The  first  law  of  thermodynamics  was  first  enunciated  by 
Mayer  in  1842,  who  obtained  a  value  for  J  from  the  dif- 
ference between  the  specific  heats  of  air  at  constant  pres- 
sure and  at  constant  volume,  which  is  equal  to  R/J  accord- 
ing to  the  next  Section.  The  value  obtained  is  not  very 
accurate  on  account  of  the  existence  of  the  Joule-Thomson 
effect  according  to  which  a  certain  amount  of  work  is  done 
in  overcoming  molecular  attraction  during  the  expansion 
of  a  gas.  But  it  would  have  been  possible  to  obtain  a  much 
better  value  by  this  method  by  using  the  gas  at  a  tempera- 
ture and  pressure  for  which  the  Joule-Thomson  effect  is 
zero;  the  method  has,  however,  now  been  superseded  by 
others.  In  1843  Joule  began  a  series  of  classical  experi- 
ments to  test  the  law  from  different  aspects,  the  best  values 
of  J  being  obtained  from  the  experiments  on  the  agitation 
of  liquids.  More  recently  J  has  been  determined  from  the 
heating  produced  by  an  electric  current.  In  the  Recuiel 
de  Constants  Physiques  (Gauthier-Villars,  1913)  published 
under  the  auspices  of  the  Societe  Frangaise  de  Physique, 
its  most  probable  value  is  taken  to  be 

J=4.184X107. 


THE  TMOLECULAR  KINETIC  AND  INTERNAL  ENERGY     39 

We  may  now  consider  more  closely  the  nature  of  heat 
capacity  of  substances. 

13.  The  Specific  Heats  of  Gases  and  Liquids. 

The  specific  heat  of  a  perfect  gas  at  constant  volume 
consists  of  the  sum  of  the  changes  in  kinetic  energy  of 
motion  of  translation  and  internal  molecular  energy  per 
degree  change  in  temperature.  If  the  specific  heat  is  ex- 
pressed in  calories,  the  first  term  is  equal  to  3R/2J  for  a 
gram  molecule  according  to  the  preceding  Section,  and 

equation  (13);    the  second    term  may  be  written    (—  ^), 

\  ol  /  v 

where  ua  denotes  the  internal  molecular  energy  per  gram 
molecule  expressed  in  terms  of  calories.  Hence  if  S'v  and 
Sv  denote  the  specific  heats  per  gram  molecule  and  per 
gram  respectively  we  have 


..  .....  (22) 

and  i 

(23) 


If  the  pressure  of  the  gas  is  kept  constant  during  the 
change  of  temperature  additional  heat  is  absorbed  in  doing 

the  external  work  ^(-^)    expressed  in   calories,  which  is 
J  \51  IP 

equal  to  R/J  according  to  equation  (4).  Therefore  if  S'P 
and  Sp  denote  the  specific  heats  at  constant  pressure  per 
gram  molecule  and  per  gram  respectively  we  have 

*--§+(!¥),  ......  ™ 

and 

S'P,     ......    .    .     (25) 


40 

where 
that 

and 


THE  MOLECULAR  CONSTANTS 

•  a  perfect  gas.    It  follows  then 


\~8TJ      VS7V 


„    R 
:,=  7, 


S'C         3R  dUg 


(26) 


(27) 


In  the  special  case  that 
becomes       =1.666. 


=0  the  foregoing  equation 


It  will  be  instructive  to  consider  the  ratio  of  the  specific 
heats  for  a  number  of  gases  at  0°  C.  given  in  Table  II.      It 

TABLE  II 


Substance. 

Formula. 

S'P 

•SV 

Mercury 

Hg 

1  666 

Argon  

A 

1.667 

Carbon  monoxide      

CO 

1  403 

Carbon  dioxide 

CO2 

1311 

Ethelene  

C2H4 

1.245 

Propane  

C3H8 

1.153 

Methyl  ether       

C2H6O 

1  107 

Ethyl  ether 

C4Hi0O 

1  097 

will  be  seen  that  for  the  mon-atomic  gases  argon  and  mercury 
the  ratio  is  practically  the  same  as  that  given  by  the  fore- 
going equation,  and  that  therefore  for  these  gases  (  — ~ )  =  0, 

i.e.,  the  kinetic  energy  of  motion  of  translation  only  changes 
with  change  of  temperature.  This  does  not,  however,  hold 


INTERNAL  ENERGY  AND  MOLECULAR  COMPLEXITY    41 

when  the  molecules  of  the  gas  consist  of  two  or  more  atoms, 
in  a  general  way  the  deviation  of  this  ratio  from  the  value 
1.666  increases  with  the  complexity  of  the  molecule.  This 
is  what  we  would  expect  since  a  change  in  temperature 
increases  the  violence  of  molecular  collision  through  the 
increase  of  velocity  of  translation,  which  would  give  rise  to 
an  increase  in  the  velocity  of  rotation  of  each  molecule, 
changes  of  the  atomic  configuration  through  collision  and 
increased  velocity  of  rotation,  etc. 

It  will  be    instructive   to  calculate  the  value  of  (—  ^) 

\°l/9 
in  the  case  of  one  of  the  complex  gases  —  say  ether.    We  have 


5R     (duA 
2/+U7V, 


and  therefore 

.  QR 
2J   ' 


fSua\      17. 
\8Tjv       2 


and  thus  the  heat  absorbed  in  changing  the  molecular 
internal  energy  is  very  much  greater  than  the  heat  3R/2J 
absorbed  in  changing  the  kinetic  energy  of  motion  of  trans- 
lation, or  in  doing  the  external  work  R/J  on  carrying 

out  the  heating  at  constant  pressure.    The  values  of  (  -^  j 

thus  vary  very  much  with  the  complexity  of  the  molecule. 

The  ratio  of  the  two  specific  heats  of  a  gas,  it  may  be 
pointed  out,  can  be  determined  with  a  much  greater  accuracy 
than  either  of  these  quantities  separately  from  measure- 
ments of  the  velocity  of  sound  Vs  which  is  given  by  the 
equation 


42         THE  MOLECULAR  CONSTANTS 

where  p  and  p  denote  the  pressure  and  density  of  the  gas. 
The  value  of  Vs  for  any  given  gas  may  be  determined  in  the 
laboratory  by  measuring  the  wave  length  X  in  a  Kundt's 
tube  corresponding  to  a  musical  note  of  known  frequency 
n,  these  quantities  being  connected  by  the  equation  Vs  =  n\. 
It  may  be  noted  that  since  sound  consists  of  the  propagation 
of  waves  of  compression  and  rarefraction,  the  velocity  with 
which  they  travel  is  closely  connected  with  the  velocity  of 
translation  of  the  molecules. 

The  specific  heat  S'a  at  constant  volume  per  gram  mole- 
cule of  a  liquid,  or  gas  not  obeying  Boyle's  law,  is  expressed 
by  the  equation 


where  Ue  denotes  the  heat  absorbed  in  internal  energy 
changes  on  allowing  the  substances  to  expand  at  constant 
temperature  till  its  volume  is  infinite.  This  equation  may 
be  proved  by  passing  the  substance  through  a  cycle  and 
equating  to  zero  the  internal  work  done.  Thus  on  expand- 
ing the  substance  at  constant  temperature  till  its  volume  is 
infinite,  and  then  raising  the  temperature  of  the  resultant 
gas  by  5T  at  constant  volume,  the  internal  work  Ue+S'v-  5T 
is  done.  On  compressing  the  substance  to  its  original  vol- 
ume at  constant  temperature,  and  lowering  its  temperature 
by  dT  at  constant  volume,  the  internal  energy  changes  by 
—  Ue—dUe—S'vr8T.  On  equating  the  sum  of  the  internal 
energies  to  zero  equation  (28)  is  obtained. 

If  the  equation  of  state  (Section  26)  of  the  substance 

under  consideration  is  known  the  value  of  I— ^j  can  at 
once  be  calculated.  According  to  thermodynamics 


«A          (6p\ 
SV)T        \STJ,-   P' 


SPECIFIC  HEATS  UNDER  VARIOUS  CONDITIONS     43 

where  U  denotes  the  total  internal  energy  of  a  substance  of 
volume  v,  temperature  T,  and  pressure  p.  On  multiplying 
this  equation  by  dv  and  integrating  it  between  the  limits 
v  and  infinity  we  obtain 


-u.-u.-r.  = 


The  right  hand  side  of  this  equation  may  be  evaluated  by 
writing  the  equation  of  state  in  the  form  p  =  <j)(T,  v)  and  sub- 
stituting for  p. 

Similarly  it  can  be  shown  that  the  specific  heat  S'Pi 
per  gram  molecule  at  constant  pressure  is  given  by 


The  internal  specific  heat  S'ipi  per  gram  molecule  at  constant 
pressure  is  therefore  given  by 


(30) 


which  is  m  times  the  specific  heat  Sg  per  gram. 

In  the  case  that  the  molecules  of  a  substance  are  partly 
dissociated  the  expressions  for  the  two  specific  heats  for  the 
gaseous  state  will  have  to  be  modified.  Consider  a  gram 
molecule  of  molecules  containing  N  molecules  of  which  Ns 
are  dissociated,  each  into  q  molecules.  The  kinetic  energy 

o  r> 

term  —  has  then  to  be  replaced  by  the  term 

3R  _ 
2j'd 

(\ 
-^)    has  to  be  replaced  by  the  term 
51  /v 

HNS\ 


44  THE  MOLECULAR  CONSTANTS 

where  H  denotes  the  heat  of  formation  of  a  gram  molecule 
of  molecules.  Equations  (22)  and  (24)  are  accordingly  re- 
placed by  the  equation 


8ua\  _Ns(8H\  _H(8NS\ 
dTj,    W\*TJ.    N\WJ,' 

and  the  equation  obtained  by  substituting  S'p,    (—  ^)   , 

\  ol  I  v 

for  Sfc,  (-TT)   in   the  foregoing  equation,  and   adding  the 


In  the  case  of  a  mixture  of  substances  it  is  useful  to 
define  the  quantity  partial  specific  heat.  Thus  if  Ser  denotes 
the  internal  heat  capacity  at  constant  pressure  of  a  mixture 
of  molecules  e  and  r,  it  may  be  written 


(32) 


where  Se  denotes  the  internal  specific  heat  associated  with 
the  molecules  e,  and  Sr  that  associated  with  the  molecules 
r.  This  equation  may  also  be  written 

Ser  =  N'rSr+N'eSe,       .....       (33) 

where  N'T  and  N'e  denote  respectively  the  numbers  of  mole- 
cules r  and  e  in  the  mixture,  and  sr  and  se  the  specific  heats 
of  a  molecule  r  and  of  a  molecule  e  respectively.  The  change 
in  heat  capacity  that  the  mixture  undergoes  on  adding  a  small 
number  of  molecules  —  say  of  r,  is  expressed  by  differentiat- 
ing the  foregoing  equation  with  respect  to  N'r,  giving 


THE  PARTIAL  SPECIFIC  HEATS  OF  MIXTURES      45 


Now  the  value  of  sr  of  a  molecule  r  depends  on  the  nature 
and  relative  positions  of  the  surrounding  molecules,  and 
the  same  holds  for  se.  These  conditions  will  evidently  be 
altered  to  a  vanishing  extent  by  the  addition  of  a  small 
number  of  molecules  r  to  the  mixture,  and  therefore  very 

(\  /     £         \ 

-r^r)     and    (-TT^-)     are   equal    to    zero. 
.ON  r/P  \ON  r/P 

The  foregoing  equation  then  gives 

t), w 

and  thus  sr  can  be  determined  by  measuring  (-^Trr)  •    The 

value  of  se  may  now  be  determined  from  equation  (33), 
or  in  the  same  way  as  sr.  In  the  latter  case  the  values  of 
sr  and  se  obtained  should  make  equation  (33)  vanish,  and 
in  this  way  therefore  the  accuracy  of  the  determinations 
may  be  tested. 

It  may  be  mentioned  here  that  the  matter  in  Sections 
30  and  39,  has  a  bearing  on  specific  heats. 

The  quantity  Ue  in  the  foregoing  equations  has  in  prac- 
tice a  finite  value  depending  on  the  temperature,  unless 
the  substance  behaves  as  a  perfect  gas,  in  which  case  the 
quantity  is  zero.  It  appears,  therefore,  that  on  changing  the 
temperature  of  a  substance  not  obeying  the  gas  laws,  a  part 
only  of  the  heat  energy  absorbed  is  converted  into  kinetic 
energy  of  translation  of  the  molecules.  The  remaining  part 
is  expended  in  other  ways,  most  probably  in  overcoming 
the  attraction  between  the  molecules,  evidence  of  the 
existence  of  which  will  now  be  considered. 

14-  Evidence  that  Molecules  and  Atoms  are  sur- 
rounded by  Fields  of  Force. 

There  is  a  good  deal  of  evidence  that  molecules  and 
atoms  are  surrounded  by  strong  fields  of  force  which  decrease 


46          THE  MOLECULAR  CONSTANTS 

rapidly  in  intensity  from  the  center  outwards.  The  internal 
heat  of  evaporation  L  of  a  liquid,  for  example,  represents 
almost  entirely  work  done  against  these  forces  during  the 
separation  of  the  molecules.  We  may  write  accordingly 


where  IJ\  —  U%  denotes  the  change  in  potential  energy  of 
the  liquid  during  evaporation  due  to  the  molecular  attrac- 
tion, and  HI  —  uu  the  change  in  internal  energy  of  the  mole- 
cules. The  free  kinetic  energy  of  the  molecules  is  not  altered 
during  the  process  of  separation  according  to  Section  21, 
and  hence  u\  —  ua  represents  the  change  in  molecular  energy 
due  to  changes  in  atomic  configuration  and  velocity  of  molec- 
ular rotation.  It  is  likely  to  be  small  in  comparison  with 
U\  —  Us,  the  work  done  against  molecular  attraction. 

When  the  volume  of  a  dense  gas  not  obeying  Boyle's 
law  is  increased  at  constant  temperature,  a  larger  amount 
of  heat  is  absorbed  than  corresponds  to  the  external  work 
done.  The  explanation  is  the  same  as  in  the  case  of  the 
evaporation  of  a  liquid. 

The  forces  in  question  are  appreciable  over  much  greater 
distances  of  separation  of  the  attracting  molecules  than  of 
the  order  of  the  distance  of  separation  in  the  liquid  state, 
i.e.,  10  ~8  cm.  This  is  shown  by  the  Joule-Thomson  effect. 
On  passing  a  stream  of  gas  at  a  volume  v  per  gram  and 
pressure  p  through  a  porous  plug  on  the  other  side  of  which 
the  gas  has  a  volume  vf  and  pressure  pf  the  temperature 
of  the  stream  emerging  from  the  plug  is  not  the  same  as  that 
entering  it.  If  the  gas  does  not  obey  Boyle's  law  p'v'  is 
not  equal  to  p  v,  and  a  part  of  the  temperature  change  is 
accounted  for  by  the  external  work  done  in  the  process, 
which  is  equal  to  p'v'  —  p  v.  This  can  evidently  at  once  be 
calculated.  The  other  part  of  the  temperature  change  is 
accounted  for  by  the  work  necessary  to  separate  the  molc- 
<  -i  ilcs  against  their  molecular  forces  during  their  passage 


THE  JOULE-THOMSON  EFFECT  47 

I 

I  through  the  porous  plug,  which  is  done  at  the  expense  of 
their  kinetic  or  heat  energy.  The  temperature  change  is 
usually  negative,  but  it  may  also  be  positive,  depending  on 
the  values  of  p,  p',  v,  and  v'.  In  the  first  case  energy  is 
required  to  separate  the  molecules,  and  hence  attraction  is 
the  paramount  force,  while  hi  the  second  case  work  is  done 
on  the  molecules,  and  the  paramount  force  is  therefore 

>  repulsion.  The  change  in  internal  molecular  energy  during 
the  process,  represented  by  u\  —  ua,  is  probably  negligible. 
Thus  we  see  that  forces  of  attraction  and  repulsion  exist 

i    between  molecules  whose  relative  magnitudes  depend  upon 

|  the  distances  of  separation;  and  the  resultant  force  may 
therefore  be  represented  by  the  sum  of  at  least  two  terms, 
one  representing  attraction  and  the  other  repulsion,  the 
value  of  the  attraction  term  decreasing  according  to  the 
nature  of  the  internal  heat  of  evaporation  of  a  liquid  and 
the  foregoing  experiments  more  rapidly  with  the  distance  of 
separation  of  the  molecules  than  the  repulsion  term. 

At  the  absolute  zero  of  temperature  the  molecules  of  a 
substance  have  no  kinetic  energy  of  translation.  It  is  fairly 
certain  that  the  volume  of  the  substance  under  those  condi- 
tions is  not  zero.  There  exists  therefore  very  probably 
another  repulsion  term  in  the  law  of  force  counteracting 
the  attraction  term  for  very  small  distances  of  separation 
of  the  molecules,  in  which  case  the  apparent  volume  of  the 
substance  would  not  vanish  at  the  absolute  zero. 

The  considerations  in  the  foregoing  paragraph  discard 
the  notion  of  molecular  volume.  It  is  important  in  this 
connection  to  note  that  the  volume  associated  with  matter 
manifests  itself  to  us  only  by  resistance  to  force;  and  mole- 
cules may  therefore  be,  and  probably  ought  to  be,  simply 
regarded  as  centers  of  force.  The  fact  that  the  apparent 
volume  of  a  molecule  depends  on  external  conditions  such 
as  the  temperature. -etc.,  appears  to  show  that  this  view 
ought  to  be  taken.  (See  Sections  19,  and  24.) 


48          THE  MOLECULAR  CONSTANTS 

It  follows  from  the  foregoing  considerations  that  as  a 
first  approximation  the  force  of  attraction  between  two 
molecules  may  be  represented  by  an  expression  consisting 
of  the  sum  of  three  terms,  one  of  which  has  a  positive  and 
the  remaining  two  negative  signs.  The  numerical  value 
of  one  of  the  negative  terms  decreases  more  rapidly  with 
the  distance  of  separation  of  the  molecules  than  the  value 
of  the  positive  term,  while  the  value  of  the  other  negative 
term  decreases  less  rapidly.  The  value  of  the  last-mentioned 
term  will  thus  be  predominant  for  large  distances  of  separa- 
tion of  the  molecules,  giving  rise  to  a  repulsion  between 
them  which  shows  itself  as  a  heating  effect  in  Joule-Thomson 
experiments.  The  value  of  the  other  negative  term  will 
be  predominant  for  close  distances  of  approach  of  the 
molecules,  giving  rise  to  a  repulsion  between  them  which 
shows  itself  as  an  apparent  volume  of  the  molecules.  For 
intermediate  distances  of  separation  of  the  molecules  the 
positive  term  will  be  predominant  and  the  force  one  of 
attraction. 

It  is  obvious  that  the  effect  of  the  molecular  forces  on 
the  properties  of  a  substance  as  a  whole  arises  through 
their  effect  on  the  relative  motion  of  the  molecules.  This 
leads  us  to  consider  the  influence  of  one  molecule  upon  the 
motion  of  another  in  a  substance. 

15.  Molecular  Interaction. 

In  treatises  on  the  Kinetic  Theory  of  Gases  molecules 
are  usually  supposed  to  consist  of  perfectly  elastic  spheres 
having  no  fields  of  force  surrounding  them.  The  interaction 
of  the  molecules  would  then  consist  simply  of  collisions. 
This  is  no  doubt  far  from  being  the  case  in  practice,  accord- 
ing to  the  previous  Section.  If  molecules  and  atoms  consist 
simply  of  centers  of  force  it  would  be  hard,  if  not  impossible, 
to  define  what  is  meant  by  a  collision.  The  mathematical 


THE  COMPLEXITY  OF  MOLECULAR  INTERACTION     49 

investigations  based  on  molecular  collision  may  therefore 
lead  to  results  having  little  to  do  with  the  actual  facts. 
It  appears  therefore  that  the  proper  development  of  the 
subject  should  be  along  lines  which  do  not  involve  molec- 
ular collision. 

This  will  be  emphasized  by  a  consideration  of  the  gen- 
eral nature  of  the  interaction  of  two  molecules  when  the  law 
of  force  of  interaction  is  of  the  nature  described  in  the  previous 
Section.  This  is  graphically  shown  in  Fig.  3,  which  in  a  gen- 
eral way  shows  the  paths  described  by  a  molecule  having  a 
high  velocity  in  approaching  a  molecule  at  rest,  correspond- 


FIG.  3. 

ing  to  different  distances  of  the  molecule  at  rest  from  the 
line  of  prolongation  of  the  initial  direction  of  the  moving  mole- 
cule. For  small  values  of  these  distances  the  path  of  the 
molecule  will  be  mainly  affected  by  the  repulsion  existing 
on  close  approach;  for  greater  distances  the  path  is  mainly 
affected  by  forces  of  attraction,  which  deflects  the  moving 
particle  towards  the  one  at  rest;  for  still  greater  distances  a 
slight  repulsion  between  the  molecules  tends  to  deflect  the 
moving  molecule  in  the  opposite  direction.  When  both 
molecules  are  moving  their  paths  are  similar  to  the  foregoing, 
the  molecules  then  simultaneously  approach  or  recede  from 
each  other,  or  move  in  the  same  direction. 

The  nature  of  the  path  of  a  molecule,  it  will  be  seen, 


50         THE  MOLECULAR  CONSTANTS 

depends  considerably  on  the  initial  direction  of  motion  of 
the  molecule,  and  in  each  case  is  complex.  In  practice 
each  path  would  probably  be  much  more  complex  than  indi- 
cated by  Fig.  3,  which  does  not  pretend  to  exhibit  the 
actual  state  of  affairs  that  exists  in  practice,  but  has  been 
drawn  to  give  the  reader  merely  some  idea  of  the  complexity 
of  the  motion  of  a  molecule  under  the  influence  of  another 
molecule.  It  is  evident,  therefore,  that  the  effect  of  the 
molecular  forces,  in  whatever  connection,  could  scarcely 
be  represented  by  the  effect  of  the  collision  of  elastic  spheres. 
In  the  case  of  a  liquid  the  paths  of  two  molecules  under 
each  other's  influence  would  be  affected  by  the  vicinity  of 
other  molecules,  and  would  therefore  be  very  much  more 
complicated  than  those  indicated  by  Fig.  3. 


CHAPTER    II 

THE  EFFECT  OF  THE  MOLECULAR  FORCES  ON  THE 
DYNAMICAL  PROPERTIES  OF  A  MOLECULE  IN  A 
DENSE  GAS  OR  LIQUID 

16.  The  Velocity  of  Translation  of  a  Molecule 
in  a  Liquid  or  Dense  Gas  when  passing  through  a 
Point  at  which  the  Forces  of  the  Surrounding  Mole- 
cules neutralize  each  other. 

It  is  often  assumed,  without  any  apparent  reason,  that 
the  average  velocity  of  a  molecule  in  a  liquid  or  dense  gas 
is  the  same  as  that  it  would  have  in  the  gaseous  state  at  the 
same  temperature.  But  this  is  not  likely  to  hold  on  account 
of  the  interaction  of  the  molecules  due  to  the  existence  of 
molecular  forces.  Information  in  this  matter  is  obtained 
from  considering  what  is  meant  by  a  thermometer  indicating 
the  temperature  of  a  gas  in  which  it  is  placed.*  From  the 
Kinetic  Theory  it  follows  that  a  thermometer  placed  in 
a  gas  assumes  its  temperature  through  being  bombarded 
by  the  gas  molecules.  The  temperature  indicated  does  not 
depend  on  the  number  of  molecules  falling  per  square  cm. 
per  second  on  the  surface  of  the  thermometer  bulb,  but 
only  on  the  average  kinetic  energy  of  a  molecule  in  the 
gas  according  to  Section  8.  The  velocity  with  which  a  mole- 
cule actually  strikes  the  bulk  depends,  however,  on  the 
attraction  exerted  by  its  material.  Therefore,  if  the  bulb 
is  covered  with  a  layer  of  material  of  much  greater  density 
*  R.  D.  Kleeman,  Phil.  Mag.,  July  1912,  pp.  100-102. 
51 


52  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

than  the  bulb  itself  the  velocity  of  impact  may  be  very  much 
increased.  But  the  temperature  indicated  by  the  ther- 
mometer is  not  altered  thereby,  as  we  know  from  experiment. 
Thus  the  temperature  indicated  corresponds  to  the  kinetic 
energy  of  a  molecule  when  not, under  the  attraction  of  the 
thermometer  bulb,  or  the  additional  velocity  given  to  the 
molecule  by  the  attraction  of  the  bulb  has  no  effect  on  the 
temperature  indicated. 

Again,  if  a  very  dense  solid  were  placed  somewhere  in 
the  gas,  the  molecules  in  its  zone  of  attraction  would  pos- 
sess a 'greater  velocity  of  translation  than  outside  the  zone. 
The  temperature  indicated  by  the  thermometer  would,  how- 
ever, not  be  altered  thereby,  but  would  correspond  to  the 
kinetic  energy  of  the  motion  of  translation  of  the  gas 
molecules  outside  this  zone,  and  the  zone  of  attraction  of 
the  thermometer  bulb,  i.e.,  to  the  kinetic  energy  when  the 
molecules  are  not  under  the  action  of  any  forces. 

The  same  result  will  also  hold  in  the  case  of  a  liquid, 
or  dense  gas,  in  which  case  the  temperature  indicated  by 
the  thermometer  corresponds  to  the  velocity  each  molecule 
has  when  not  under  the  action  of  a  force,  which  occurs  when 
passing  through  a  point  where  the  forces  of  the  surrounding 
molecules  neutralize  each  other.  There  are  evidently  num- 
bers of  such  points  in  a  liquid  which  change  their  positions 
with  the  motion  of  the  molecules.  The  velocity  which  a 
molecule  has  when  passing  through  such  a  point  is  not 
necessarily  always  the  same,  but  an  average  velocity  may 
be  associated  with  it.  When  the  molecule  is  not  passing 
through  such  a  point  hi  a  dense  gas  or  liquid  it  is  under  the 
action  of  forces,  and  its  velocity  on  the  average  is  then 
greater  (Section  17)  than  the  foregoing  average  velocity. 
Therefore  the  average  velocity  of  a  molecule  when  not 
under  the  action  of  forces  may  be  called  its  average  minimum 
velocity. 

It  is  evident  then  that  if  we  consider  two  masses  of  the 


THE  AVERAGE  MINIMUM  MOLECULAR  VELOCITY     53 

same  substance,  one  in  the  perfectly  gaseous  state  and  the 
other  in  the  liquid  state,  and  they  possess  the  same  tem- 
perature, the  average  kinetic  energy  of  a  gas  molecule  is 
the  same  as  the  average  minimum  kinetic  energy  of  a  mole- 
cule in  the  liquid  state,  in  other  words,  the  average  kinetic 
energy  of  a  molecule  when  not  under  the  action  of  a  force  in 
a  liquid  or  dense  gas  is  the  same  as  the  average  kinetic  energy 
it  would  have  in  the  perfect  gaseous  state  at  the  same  tem- 
perature. 

We  may  also  reason  in  a  reverse  manner.  A  thermometer 
placed  in  a  liquid  indicates  its  temperature  through  being 
bombarded  by  the  liquid  molecules.  The  velocity  of  impact 
and  the  general  nature  of  the  bombardment  may  be  changed 
without  changing  the  temperature  indicated  by  changing 
the  nature  of  the  thermometer  bulb,  or  compressing  the 
liquid  at  constant  temperature.  The  question  then  arises: 
what  is  the  connection  between  the  temperature  indicated 
and  the  velocity  of  translation  of  a  molecule  under  stated 
conditions  in  the  liquid?  Now  the  velocity  of  a  molecule 
that  means  anything  definite  in  a  liquid  corresponds  to  its 
independent  velocity,  or  the  velocity  when  not  under  the 
action  of  an  external  force,  and  we  must  therefore  endeavor 
to  connect  this  velocity  with  the  temperature  indicated. 
Now  in  the  case  of  a  gas  a  thermometer  indicates  the  tem- 
perature corresponding  to  the  kinetic  energy  of  a  molecule 
when  not  under  the  action  of  a  force.  There  is  no  reason 
whatever  why  this  should  not  hold  in  the  case  of  a  liquid. 
We  conclude,  therefore,  that  the  average  kinetic  energy  of  a 
molecule  in  a  liquid  under  these  conditions  is  the  same  as 
in  the  gaseous  state  at  the  same  temperature. 

This  result  may  be  established  in  a  somewhat  different 
way.  Consider  two  chambers  having  a  wall  in  common  filled 
with  gases  at  the  same  temperature.  Suppose  that  one-half 
of  the  common  wall  adjacent  to  one  of  the  gases  is  replaced 
by  a  much  denser  material  so  that  the  molecules  strike  the 


54          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

wall  on  that  side  with  a  greater  velocity  than  on  the  other 
side.  This  change  in  the  nature  of  one  side  of  the  wall  will 
not  give  rise  to  a  flow  of  heat  from  one  side  to  the  other, 
since  this  would  be  contrary  to  the  laws  of  thermodynamics. 
The  heat  effect  produced  by  the  molecules  on  impinging 
on  the  wall  therefore  does  not  depend  on  the  velocity  of 
impact,  but  on  the  velocity  when  not  under  the  action  of  the 
attraction  of  the  wall.  The  foregoing  result  follows  then  in 
a  similar  way  as  before. 

We  will  now  turn  our  attention  to  the  velocity  of  a  mole- 
cule at  other  points  in  a  liquid  than  those  considered. 

17.  The  Total  Average  Velocity  of  Translation 
of  a  Molecule  in  a  Substance. 

When  a  molecule  is  not  passing  through  a  point  in  a 
dense  substance  at  which  the  forces  of  the  surrounding 
molecules  neutralize  each  other,  its  velocity  is  likely  to 
differ,  on  account  of  the  effect  of  molecular  attraction  and 
repulsion,  from  its  velocity  at  such  a  point,  which  we  have 
seen  in  the  previous  Section  is  the  same  as  that  it  would 
have  in  the  perfectly  gaseous  state  at  the  same  temperature. 
The  total  average  velocity  of  the  molecule,  corresponding  to 
the  complete  path  described  during  an  infinitely  long  time, 
is  therefore  likely  to  differ  from  the  velocity  defined  as 
the  average  minimum  velocity.  When  a  substance  is  in  the 
perfectly  gaseous  state  these  two  velocities  are  evidently 
very  approximately  equal  to  one  another,  or  denote  approxi- 
mately the  same  thing.  But  evidently  this  cannot  be  the 
case  when  the  density  of  the  substance  is  such  that  the  motion 
of  the  molecule  is  constantly  influenced  by  the  attraction 
and  repulsion  of  the  surrounding  molecules.  In  fact,  it 
can  be  shown*  that  in  the  case  of  a  liquid  the  total  average 

*  R.  D.  Kleeman,  Phil.  Mag.,  July,  1912,  pp.  101-103. 


THE  TOTAL  AVERAGE  MOLECULAR  VELOCITY      55 

velocity  is  several  times  that  of  the  average  minimum 
velocity. 

Suppose  that  the  volume  of  a  gram  of  liquid  is  changed 
by  a  small  amount  at  a  low  temperature  in  contact  with  its 
saturated  vapor  by  changing  the  temperature.  The  energy 
spent  per  molecule  in  overcoming  the  molecular  attraction 
is  equal  to  ma-dL,  where  L  denotes  the  internal  heat  of 
evaporation  in  ergs  per  gram  and  ma  the  absolute  molecular 
weight  of  a  molecule.  The  average  force  acting  on  a  mole- 
cule during  the  expansion  is  therefore  -i-wa,  where  x  denotes 

dx 

the  distance  of  separation  of  the  molecules.  Now  x  =  (  —  j  , 
where  p  denotes  the  density  of  the  liquid,  and  the  force  may 

therefore  be  written  3raa2/3p4/3-r~-    We  may  assume  without 

dp 

any  serious  error  that  on  the  average  the  expenditure  of 
energy  on  a  molecule  during  its  motion  of  translation  is 
proportional  to  the  distance  traversed,  and  the  force  acting 
equal  to  the  foregoing  value.  Therefore,  when  a  molecule 
traverses  a  distance  equal  to  half  the  distance  of  separation 
of  the  molecules  it  may  on  the  average  gain  or  lose  the 

amount  of  energy  -raap— ,  which  corresponds  to  a  change 
Z       dp 

in  velocity  in  cms.  equal  to  A/3p-r-.    For  example,  in  the 

casfe  of  ether  at  0°  C,  P  =  .7362,   aid  g.  2.72*118X10* 

dp  .UU14 

corresponding  to  a  change  of  10°  C.,  which  gives  a  change  in 
velocity  equal  to  1.69  X105  cms. /sec.  of  the  molecule.  The 
average  minimum  velocity  of  a  molecule,  which  corresponds 
to  that  of  a  molecule  in  the  gaseous  state  at  the  same  tem- 
perature, is  equal  to  3.03  XlO4  cms. /sec.  We  see  therefore 
that  the  total  average  velocity  of  a  molecule  in  a  liquid  may 
be  several  times  that  corresponding  to  its  temperature. 


56  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

It  will  be  convenient  now  to  connect  this  velocity  by  a 
general  equation  with  a  quantity  which  has  already  been 
discussed  in  connection  with  gases. 

18.  The  Number  of  Molecules  crossing  a  Square 
Cm.  in  a  Substance  of  any  Density  from  one  Side  to 
the  Other  per  Second. 

If  the  molecules  consisted  in  three  equal  streams  moving 
at  right  angles  to  each  other,  and  a  plane  one  square  cm. 
in  area  were  taken  at  right  angles  to  one  of  the  streams, 
NcVt  (2  X  3)  molecules  would  cross  it  in  one  direction  per  sec- 
ond, and  the  same  number  in  the  opposite  direction,  where 
Nc  denotes  the  number  of  molecules  per  cubic  cm.,  and  Vt 
the  total  average  velocity.  This  expression  may  be  obtained 
and  along  the  same  lines  as  the  similar  expression  in  equa- 
tion (17).  If,  however,  the  molecules  move  in  all  directions, 
as  is  actually  the  case,  the  number  of  molecules  n  crossing 
a  square  cm.  from  one  side  to  the  other  in  all  directions  is 
double  the  above  number,  that  is 

NcVt 
rc  =  —  ^-.       ......     (35) 

This  may  be  proved  by  means  of  the  geometrical  considera- 
tions used  in  Section  11.  The  result  may,  however,  be  at 
once  deduced  from  the  results  obtained  in  that  Section. 
Thus  it  was  shown  that  the  pressure  of  a  gas  may  be  written 

A 


where  n0  denotes,  if  the  molecules  consist  of  three  equal 
streams  at  right  angles,  the  number  crossing  a  square  cm. 
taken  at  right  angles  to  one  of  the  streams,  and  n  denotes 
the  number  of  molecules  when  they  move  in  all  directions, 
as  is  the  case  in  practice.  Hence  it  follows  that  n  = 


THE  INTERNAL  MOLECULAR  VOLUME  OF  A  GAS     57 

the  result  just  used,  which  obviously  does  not  hold  only  for 
a  gas,  but  in  general. 

A  molecule  probably  does  not  consist  of  a  point  in  space, 
but  possesses  a  real  or  apparent  molecular  volume,  which 
influences  its  motion  and  consequently  the  properties  of  the 
substance  of  which  it  forms  part.  This  effect  will  be  first 
considered  in  connection  with  a  gas. 

19.   The  Effect  of  Molecular  Volume  on  p  and 
n  in  the  Case  of  an  Imperfect  Gas. 

We  have  seen  in  Section  6  that  the  effect  of  the  volume 
of  the  molecules  of  a  substance  is  to  increase  its  external 
pressure  from  that  it  would  have  if  the  molecules  possessed 
no  volume.  The  effect  of  a  molecular  volume  is  evidently 
to  decrease  the  space  available  for  molecular  motion  of 
translation.  It  is  therefore  equivalent  to  supposing  that 
the  molecules  are  devoid  of  volume,  and  that  the  volume 
of  the  substance  (as  a  whole)  is  decreased  by  an  amount 
b,  as  is  shown  by  means  of  a  diagram  in  Fig.  4.  This  quan- 
tity is  the  apparent  volume  of  the  molecules  connected  with 
molecular  motion.  The  external  pressure  p  in  this  case 
might  thus  be  written 


in  conformity  with  the  equation  p  =  RT/v  of  a  perfect  gas, 
where  v  refers  to  a  gram  molecule  of  molecules. 

It  will  not  be  difficult  to  see  that  this  result  will  also 
hold  if  the  apparent  volume  of  the  molecules  is  caused  by 
molecular  forces  of  repulsion,  which  do  not  permit  an  ap- 
proach of  two  molecules  beyond  a  certain  distance. 

If  the  molecules  of  a  gas  consist  of  a  number  of  hard 
elastic  spheres  not  surrounded  by  fields  of  force,  their  vol- 
ume would  not  interfere  with  their  velocity  of  translation, 


53 


THE  EFFECT  OF  THE  MOLECULAR  FORCES 


which  would  simply  correspond  to  the  temperature  of  the 
gas.  This  will  be  made  clear  by  an  inspection  of  Fig.  4 
according  to  which  a  set  of  molecules  possessing  molecular 
volume  of  the  foregoing  kind  move  in  the  space  v  with  the 
same  velocity  as  if  they  were  devoid  of  volume  and  moved 
in  the  space  v— b.  Equality  of  velocities  is  necessary  in 
the  two  cases  since  the  external  pressures  are  the  same. 
Since  the  molecular  forces  may  give  rise  to  an  apparent 
volume  of  the  molecules  as  well  as  affect  their  velocity  of 


FIG.  4. 

translation,  the  foregoing  considerations  suggest  a  defini- 
tion of  molecular  volume  which  is  quite  definite  and  applies 
to  all  states  of  matter.  Thus  the  apparent  molecular  volume 
of  a  substance  may  be  defined  as  the  quantity  associated 
with  the  molecules  whose  magnitude  affects  the  external 
pressure  of  the  substance  at  constant  -temperature  and  vol- 
ume, but  does  not  affect  the  average  velocity  of  translation 
of  the  molecules.  The  latter  property  is  expressed  by  the 
equation 


THE  MOLECULAR  VOLUME  AND  VELOCITY         59 

where  Vt  denotes  the  average  velocity  of  translation,  and 
b  the  apparent  volume  of  the  molecules.  The  velocity 
Vt  is  a  function  of  the  temperature  and  therefore 

5VA       /SVA        / 56  \       /SFA     _  /SVA 
P/i     \8b/T,v    (dT/^ \5T)6.,~\8T)*t,' 

by  the  preceding  equation.  It  is  also  a  function  of  the 
volume  and  therefore 

/5VA       /5FA       /«6\       /SFA      =  /*FA 
V  to  /r     \dbJT,v   \dv/  T     \5v  J T,b     \dv  J Tti>' 

These  equations  express  the  relations  between  Vt  and  6 
according  to  the  above  definition  of  b. 

It  should  be  carefully  noted  that  6  is  a  perfectly  definite 
mathematical  quantity,  though  there  might  be  some  diffi- 
culties in  defining  the  exact  nature  and  geometrical  con- 
figuration of  the  real,  or  apparent,  volume  of  a  single  mole- 
cule. 

The  apparent  volume  of  a  molecule,  whatever  its  cause, 
is  likely  to  change  little  if  at  all  with  the  density  of  the 
substance.  The  value  of  6  may  therefore  without  the  risk 
of  introducing  any  serious  error  be  taken  constant  over  small 
changes  in  density.  This  is  very  useful  in  the  determina- 
tion of  the  values  of  6  from  simultaneous  equations,  as  is 
carried  out  in  Section  29. 

The  value  of  6  would  vary  with  the  temperature  if  caused 
by  forces  of  repulsion  between  the  molecules.  Thus  two 
molecules  moving  towards  each  other  in  a  substance  along 
the  same  straight  line  continue  approaching  each  other 
until  their  kinetic  energy  of  translation  is  completely  trans- 
formed into  potential  energy  of  repulsion,  after  which  they 
retrace  their  path.  Hence  a  nearer  approach  must  take 
place  with  an  increase  of  temperature,  which  corresponds 
to  an  increase  in  kinetic  energy,  and  a  corresponding  decrease 
in  the  value  of  b  would  result. 


60          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

It  will  be  evident  on  reflection  that  the  forces  of  attrac- 
tion and  repulsion  between  two  molecules  might  be  changed 
at  constant  temperature  and  volume  so  that  Vt  remains 
the  same,  in  which  case  a  change  in  b  may  result,  or  so  changed 
that  6  remains  the  same,  in  which  case  a  change  in  Vt  may 
result.  This  shows  that  the  definition  of  b  given  is  admissible. 

A  change  in  temperature  or  density  of  a  substance  may 
evidently  result  in  a  change  of  both  b  and  Vt.  The  fore- 
going equations  then  express  that  whether  or  no  b  changes, 
the  changes  in  Vt  are  the  same  as  if  b  remained  constant. 

It  can  be  shown  that  n,  the  number  of  molecules  passing 
through  a  square  cm.  in  one  direction  per  second,  is  inde- 
pendent of  the  apparent  volume  of  the  molecules,  since  the 
velocity  of  translation  is  also  independent  of  this  quantity. 
Let  us  suppose  that  we  are  initially  dealing  with  molecules 
devoid  of  volume,  and  to  which  is  then  given  an  apparent 
volume  b.  We  may  then  suppose  that  the  resultant  mole- 
cules possess  no  volume  and  occupy  the  volume  v  —  b  of 
the  volume  v  for  motion  of  translation.  If  Nc  denote  the 
number  of  molecules  per  cubic  cm.  when  the  gas  occupies 
the  volume  v,  the  number  when  it  occupies  the  volume  v—b 
is  Ncv/(v—b).  The  number  of  molecules  crossing  a  square 
cm.  per  second  is  accordingly  changed  from  n  to  nv/(v  —  b), 
since  the  velocity  of  translation  remains  unaltered,  where 
n  corresponds  to  the  molecules  having  no  volume  and 
occupying  as  a  whole  the  volume  v.  But  in  practice  the 
molecules  with  their  apparent  volumes  would  be  distributed 
throughout  the  volume  v,  instead  of  being  separated  from 
their  volumes  as  shown  in  Fig.  4,  and  the  number  of  mole- 
cules crossing  a  square  cm.  per  second  is  therefore  equal  to 
the  foregoing  expression  reduced  in  the  ratio  of  v  —  b  to  v, 
which  makes  it  equal  to  n,  the  value  when  the  molecules 
have  no  volume. 

This  result  can  be  very  simply  deduced  from  equation 
(35).  If  the  apparent  volume  b  of  the  molecules  is  changed 


THE  MOLECULAR  VOLUME  AND  GAS  PRESSURE     61 

without  changing  Vt,  by  definition  n  remains  constant 
according  to  this  equation,  and  it  is  thus  independent  of 
the  volume  of  the  molecules.  This  method  of  obtaining 
the  result  is  perhaps  not  so  instructive  as  the  preceding 
one.  The  result  is  mathematically  expressed  by  the  equation 


(£)  =°- 

\db/  T,V 


By  means  of  this  equation  and  the  Differential  Calculus 
it  can  be  shown,  similarly  as  in  connection  with  the  quan- 
tity Vt)  that 

5n\       /5n\  /dn\        S5n 

and 


These  equations   express  the  relations  between  n  and   6 
according  to  its  definition. 

If  the  molecules  of  a  gas  were  devoid  of  volume  and  molec- 
ular forces  the  laws  of  a  perfect  gas  apply,  and  the  external 
pressure  p  of  the  gas  may  then  be  written 


according  to  equation  (20)  in  Section  11. 

The  acquisition  of  an  apparent  volume  b  by  the  mole- 
cules does  not  change  n,  we  have  seen,  but  it  changes  the 
external  pressure  in  the  ratio  of  v  to  v—  b.  Hence  the  external 
pressure  of  a  gas  under  these  conditions  may  be  written 


Since  n  is  defined  and  used  in  connection  with  the  pres- 
sure produced  by  the  molecules  of  a  substance,  the  crossing 
of  a  plane  by  a  molecule,  if  the  molecules  are  devoid  of 
volume,  is  obviously  associated  with  the  crossing  of  its 


62          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

center  of  mass.  This  also  holds  when  the  molecules  possess 
volume,  according  to  the  method  the  subject  has  been 
developed. 

The  external  pressure  of  a  gas  is  equal  to  the  expansion 
pressure  due  to  the  motion  of  translation  of  the  molecules 
tending  to  expand  the  gas.  But  in  the  case  of  a  liquid  or 
dense  gas  this  does  not  hold,  due  to  the  effect  of  the  forces 
of  attraction  and  repulsion  between  the  molecules,  and  the 
expansion  pressure  of  a  dense  substance  is  therefore  a  quan- 
tity requiring  special  consideration. 

£0.  The  Expansion  Pressure  exerted  by  the  Mole- 
cules of  a  Substance  of  any  Density. 

The  pressure  exerted  by  a  gas  in  a  closed  vessel  upon  its 
walls  is,  according  to  the  Kinetic  Theory  of  Gases,  due  to  the 
change  in  momentum  the  molecules  undergo  through  col- 
lision with  the  vessel's  walls.  The  velocity  of  translation 
of  the  molecules,  the  density  and  external  pressure  of  the 
gas,  are  then  connected  by  equation  (7)  in  Section  6.  This 
equation  holds,  however,  strictly  only  if  the  walls  of  the 
vessel  do  not  exert  any  attraction  upon  the  molecules.  It 
holds  very  approximately  in  the  cases  usually  occurring  in 
practice,  in  which  the  size  of  the  vessel  is  so  large  that  the 
greater  mass  of  the  gas  is  only  slightly  under  the  influence 
of  the  attraction  of  the  walls.  But  under  certain  conditions 
this  influence  may  be  so  large  that  the  equation  cited  does 
not  hold.  It  will  be  of  importance  to  investigate  this  effect 
of  molecular  forces  more  closely,  as  it  has  a  bearing  on  the 
kinetic  properties  of  liquids.  Thus  let  AB  and  CD,  in  Fig. 
5,  represent  two  opposite  walls  of  a  vessel,  and  let  the  planes 
A'B'  and  C'D'  denote  the  boundaries  of  the  zones  in  which 
the  attraction  of  the  walls  is  of  appreciable  magnitude. 
A  molecule  on  entering  one  of  the  zones  has  its  velocity 
increased  through  the  attraction  of  the  material  of  tt$ 


THE  EXPANSION  PRESSURE  EQUATION 


63 


adjacent  wall.  Thus  the  molecule  would  exert  a  pull  upon 
the  wall  during  the  whole  time  it  is  in  the  zone.  Now  it 
follows  from  the  dynamical  equation  Ft  =  maV  that  the  aver- 
age pull  per  unit  time  exerted  by  the  molecule  during  the 
time  it  is  in  the  zone,  is  balanced  by  the  average  thrust 
acting  in  the  opposite  direction  upon  the  wall  during  that 
period,  due  to  the  reversal  of  the  direction  of  the  additional 
momentum  acquired  by  the  molecule  in  the  zone.  The 
force  exerted  by  the  molecule  upon  the  wall  during  one 
rebound  is  thus  not  affected  by  molecular  attraction.  The 


B 


FIG.  5. 


total  pressure  exerted  by  the  molecules  of  the  vessel  on  the 
wall  would  therefore  depend  only  on  the  number  impinging 
per  square  cm.  per  second  on  the  wall.  This  number  is 
equal  to  the  number  of  molecules  entering  a  zone  per  square 
cm.  per  second.  It  evidently  depends  upon  the  velocity 
of  translation  of  the  molecules  outside  the  zones,  and  the 
number  of  molecules  per  cubic  cm.  in  that  region.  If  the 
thickness  of  each  zone  is  small  in  comparison  with  the 
diameter  of  the  vessel,  the  average  velocity  of  translation 
of  the  molecules  may  be  taken  equal  to  their  velocity  out- 
side their  zones. 

If,  however,  the  thickness  of  the  zone  is  comparable 
with  the  diameter  of  the  vessel  this  no  longer  holds.    Thus 


64 


THE  EFFECT  OF  THE  MOLECULAR  FORCES 


suppose  that  the  walls  AB  and  CD  are  so  close  to  one  another 
that  the  zones  of  attraction  of  the  walls  overlap  as  is  shown 
in  Fig.  6.  The  attraction  exerted  by  one  of  the  walls  would 
be  neutralized  by  that  of  the  other  wall  in  a  plane  EF  which 
lies  midway  between  the  walls.  The  velocity  of  a  molecule 
in  that  plane  would  therefore  be  the  same  as  that  it  would 
have  between  the  planes  A'B'  and  C'D'  in  Fig.  5.  For 
according  to  Section  16  the  velocity  of  translation  of  a  mole- 
cule at  a  point  where  the  various  forces  neutralize  each 

other,  is,  at  constant  tempera- 
ture, the  same  as  its  velocity 
when  under  the  action  of  no 
force.  The  velocity  for  all  points 
outside  of  the  plane  EF,  would, 
however,  be  considerably  greater 
than  that  in  the  plane.  The 
latter  velocity  would  therefore 
be  considerably  smaller  than 
the  total  average  velocity.  As 
before  the  attraction  of  the 
walls  would  not  affect  the  force 
exerted  by  a  molecule  corresponding  to  a  single  rebound. 
But  it  would  evidently  affect  the  number  of  times  a 
molecule  crosses  from  one  wall  to  the  other  per  second. 
Therefore  the  pressure  the  molecules  would  exert  upon  the 
walls  AB  and  CD  under  the  conditions  shown  in  Fig.  6 
would  indirectly  be  increased  by  the  molecular  attraction 
of  the  walls.  The  pressure  per  square  cm.  on  each  wall 
would  accordingly  be  equal  to  nA/2,  where  n  denotes  the 
number  of  molecules  crossing  a  square  cm.  per  second  of 
the  plane  EF  from  one  side  to  the  other — this  quantity 
being  affected  by  the  attraction  of  the  walls  on  the  mole- 
cules, and  A/2  denotes  the  force  exerted  by  a  single  mole- 
cule on  rebounding  from  one  of  the  walls,  this  quantity 
having  the  same  value  as  when  the  walls  possess  no  attrac- 


i 
i 
i 

F 

FIG.  6. 


THE  EXPANSION  PRESSURE  EQUATION  65 

tion,  its  value  having  been  determined  in  Section  11.  A 
similar  state  of  affairs  exists  in  a  liquid,  or  in  a  substance 
which  does  not  behave  as  a  perfect  gas,  the  attraction  be- 
tween the  molecules  taking  the  place  of  the  attraction  of 
the  foregoing  walls. 

These  deductions  will  now  be  used  to  find  an  expression 
for  the  expansion  pressure  Pe  of  a  substance  tending  to 
expand  it,  due  to  the  motion. of  translation  of  the  mole- 
cules. The  number  of  molecules  n  crossing  a  plane  one 
square  cm.  in  area  in  all  directions  from  one  side  to  the 
other,  is  equal  to  the  number  crossing  in  the  opposite  direc- 
tion. We  may  therefore  suppose,  if  the  molecules  possess 
no  volume,  that  each  stream  of  molecules  is  reflected  from 
the  foregoing  plane  and  thus  exerts  a  pressure  upon  it. 
Hence  the  expansion  pressure  Pe  would  be  equal  to  the 
number  of  molecules  crossing  the  plane  in  one  direction 
multiplied  by  the  factor  A/2,  which  expresses  the  force 
exerted  by  a  single  molecule,  which  factor  is  the  same  as 
applies  in  the  case  of  a  perfect  gas  according  to  the  pre- 
ceding investigation,  that  is,  under  these  conditions, 


But  if  each  molecule  has  an  apparent  volume  it  may  exert 
a  pressure  across  the  plane  without  crossing  it,  and  the  fore- 
going formula  has  to  be  modified  accordingly.  We  have 
seen  in  Section  19  that  the  molecular  volume  does  not  affect 
the  value  of  n,  but  increases  the  expansion  pressure  in  the 
ratio  of  v  to  v— b.  Hence  in  general  the  expansion  pressure 
is  given  by 

4  =  n  -^2.543  X  W-20\/Tm,         (38) 


v-b  2 
where  v  denotes   the  volume  of  a   gram   molecule,  and  b 


66  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

the  molecular    volume,  the  value   of    A    being   given   by 
equation  (19).     This  equation  may  be  written 


(39) 


by  means  of  equation   (35),  where   Vt  denotes    the   total 
average  velocity  of  a  molecule,  and  Nc  the  molecular  con- 

centration.    Since   V=  J—  ,   and   ^  =  maj—       (Sec- 
\    m  2  \   m 

tions  6  and  11),  the  foregoing  equation  may  be  written  by 
the  help  of  the  equation  vNcma  =  m, 


where  V  denotes  the  average  kinetic  energy  velocity  of  a 
molecule,  and  b  is  taken  to  refer  to  a  gram  molecule  01 
molecules. 

The  foregoing  equations  are  not  corrected  for  the  dis- 
tribution of  molecular  velocities  in  the  gaseous  state,  since 
the  function  A  in  equation  (38)  ,  from  which  the  other  equa- 
tions are  derived,  has  not  been  thus  corrected.  If  this  cor- 
rection is  carried  out  according  to  Maxwell's  law  the  right 


hand  side  of  each  equation  has  to  be  multiplied  by  (  +  1—  j, 

or  1.085,  according  to  Section  11.  It  may  be  noted  that  if 
the  kinetic  energy  velocity  V  of  a  molecule  in  the  gaseous 
state  is  eliminated  from  equation  (40)  by  means  of  the 
equation  Fa=.9227  given  in  Section  8,  and  the  equation 
is  corrected  for  Maxwell's  law,  an  equation  is  obtained  in 
which  V  is  replaced  by  Va  the  total  average  velocity  of  a 
molecule  in  the  gaseous  state.  The  equation  in  this  form 
is  independent  of  the  distribution  of  molecular  velocities 
when  applied  to  the  gaseous  state,  since  then  Vt  =  Va. 
If  a  substance  behaved  as  a  perfect  gas  at  all  densities 


THE  EXPANSION  PRESSURE  OF  MIXTURES  67 

we  would  have  Vt=Va  and  6  =  0.  The  expansion  pressure 
is  then  equal  to  the  external  pressure,  according  to  equa- 
tion (40)  in  the  corrected  form  where  Va  takes  the  place  of 
V,  and  the  gas  equation.  But  since  this  is  not  the  case 
in  practice,  for  Vt>Va  according  to  Section  17,  and  b  is 
not  zero  according  to  Section  29,  the  expansion  pressure 
is  greater  than  the  pressure  p  =  RT/v.  Hence  there  must 
exist  a  negative  pressure  acting  in  the  opposite  direction 
to  the  external  pressure  to  ensure  equilibrium  of  the  sub- 
stance, since  in  practice  the  external  pressure  is  usually 
less  than  RT/v.  This  negative  pressure  is  called  the  intrin- 
sic pressure  and  is  discussed  in  the  next  Section. 

The  foregoing  results  may  be  extended  to  a  mixture  of 
molecules.  Let  us  consider  a  mixture  of  two  different  mole- 
cules e  and  r.  Since  the  expansion  pressure  of  a  substance 
is  caused  by  the  motion  of  translation  of  the  molecules, 
and  each  molecule  produces  pressure  independently,  the 
total  expansion  pressure  is  equal  to  the  sum  of  the  expan- 
sion pressures  exerted  by  the  molecules  e  and  r.  The  expan- 
sion pressure  exerted  by  the  molecules  e  is  evidently  equal  to 


Ve  — 

where  ne  denotes  similarly  as  before  the  number  of 
molecules  e  crossing  a  square  cm.  from  one  side  to  the 
other  per  second,  me  denotes  the  relative  molecular  weight 
of  a  molecule,  ve  may  be  taken  to  refer  to  the  volume  of  the 
mixture  containing  a  gram  molecule  of  molecules  e,  and 
b'e  denotes  the  apparent  volume  in  the  volume  vc  which 
the  molecules  e  and  r  appear  to  possess  in  obstructing  the 
motion  of  a  molecule  e.  Similarly  the  expansion  pressure 
exerted  by  the  molecules  r  is  equal  to 

nr  —jT-2.543  X  10-  20VTmr, 


68          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

where  the  symbols,  vr,  nr,  mr,  and  b'r,  have  meanings  similar 
to  the  symbols,  ve,  ne,  me,  and  b'e.  The  total  expansion  pres- 
sure is  therefore  given  by 

(41) 

and  its  value  in  the  case  of  a  mixture  of  any  number  of 
different  kinds  of  molecules  is  therefore  given  by 


.     .     (42) 

Similarly  corresponding  to  equations  (39)  and  (40)  we  have 

.     .     .     (43) 


and 

"  ......  <«> 


for  any  number  of  constituents  in  the  mixture.  The  fore- 
going equations  may  be  corrected  for  Maxwell's  distri- 
bution law  similarly  as  before. 

Approximate  values  of  b'e  and  b'T  in  equation  (41)  may 
be  obtained  from  the  values  of  be  and  br  referring  to  the 
constituents  in  the  pure  state.  The  value  of  b'e  arises  from 
the  interaction  of  molecules  e  with  each  other  and  with 
the  molecules  r,  while  the  value  of  b'r  arises  from  the  inter- 
action of  the  molecules  r  with  each  other  and  with  the  mole- 
cules e.  It  is  evident,  therefore,  that  if  Nr  denotes  the 
concentration  of  the  molecules  r,  and  ve  the  volume  of  the 
mixture  containing  a  gram  molecule  of  molecules  e,  we  have 
approximately 


where  N  denotes  the  number  of  molecules  in  a  gram  mole- 
cule of  a  pure  substance.    The  first  term  on  the  right-hand 


THE  MOLECULAR  VOLUME  OF  MIXTURES 


69 


side  of  the  equation  expresses  the  apparent  molecular 
volume  of  the  molecules  e  when  a  molecule  e  interacts  with 
them,  while  the  second  term  expresses  the  apparent  volume 
of  the  molecules  r  when  a  molecule  e  interacts  with  them. 
Similarly  for  b'r  we  have  approximately 


"N 


I 


where  Ne  denotes  the  number  of  molecules  e  in  a  volume 
of  the  mixture  containing  a  gram  molecule  of  molecules  r. 

The  expansion  pressure  Pe  may  now  be  connected  with 
other  quantities. 

21.  The  Intrinsic  Pressure;  and  the  Equation  of 
Equilibrium  of  a  Substance. 

On  account  of  the  attraction  between  the  molecules  of 
a  liquid,  or  dense  gas,  a  negative  pressure  is  associated 
with  it  usually  called  the  intrinsic  pressure,  which  acts  in 


B 


FIG.  7. 

the  opposite  direction  to  the  external  pressure  regarded 
from  an  external  point.  The  existence  of  this  pressure 
follows  from  the  following  consideration.  Suppose  a  sub- 
stance is  cut  into  two  parts  A  and  B  by  an  imaginary 
plane  ab  as  shown  in  Fig.  7.  If  attraction  between  the 
molecules  exists  the  parts  A  and  B  exert  an  attraction 


70          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

upon  each  other  which  gives  rise  to  a  pressure  between 
them.  This  pressure  per  square  cm.  in  the  plane  ab  is 
numerically  equal  to  the  intrinsic  pressure,  or  negative 
pressure  which  the  parts  A  and  B  exert  upon  each  other. 

The  expansion  pressure  Pe  exerted  by  the  molecules  in 
crossing  a  square  cm.  in  the  plane  ab  is  evidently  balanced 
by  the  intrinsic  pressure,  which  will  be  denoted  by  P», 
and  the  external  pressure  p,  that  is, 

Pn+P  =  Pe  .......       (45) 

or 


according  to  equation  (38).  The  foregoing  equation* 
is  the  equation  of  equilibrium  of  a  substance.  The  quan- 
tity Pn  may  be  expressed  very  approximately  in  terms  of 
quantities  which  may  be  measured  directly,  as  will  now 
be  shown. 

The  average  position  of  a  molecule  in  a  substance  cor- 
responds to  a  point  about  which  the  surrounding  mole- 
cules are  symmetrically  situated.  The  molecular  forces 
therefore  neutralize  each  other  at  such  a  point.  The  aver- 
age kinetic  energy  of  a  molecule  at  its  average  position  is 
therefore  the  same  as  that  it  has  in  the  perfectly  gaseous 
state  according  to  Section  16.  Now  we  may  suppose  that 
each  molecule  occupies  its  average  position  in  a  substance 
at  the  same  time  for  purposes  of  calculation,  for  the  prop- 
erties of  a  molecule  in  connection  with  its  forming  part  of 
a  substance  are  average  properties.  It  appears  then  that 
if  the  volume  of  a  substance  is  changed  by  8v  at  constant 
temperature,  the  average  minimum  or  temperature  kinetic 
energy  is  not  changed.  Hence  the  heat  5U  absorbed  repre- 

*  Reconstructed  form  of  an  equation  given  by  the  writer  in  the 
Phil.  Mag.,  July,  1912,  pp.  101-108. 


THE  INTRINSIC  PRESSURE  EQUATION  71 

sents,  if  np  dissociation  occurs  —  as  we  will  suppose,  the 
work  dUa  done  against  molecular  attraction  in  separating 
the  molecules,  and  in  changing  their  internal  molecular 
energy  by  8um.  The  change  in  energy  dum  may  be  caused 
by  a  change  in  the  atomic  configuration  of  the  molecules, 
and  a  change  in  their  velocities  of  rotation,  etc.  The  latter 
energy  is  very  likely  small  in  comparison  with  the  former, 
in  which  case  we  have 

Pn-5v=dUa=dU, 

since  work  is  done  against  the  intrinsic  pressure  during 
expansion.  Now  according  to  thermodynamics  we  have 


where  p  and  T  denote  the  pressure  and  absolute  tempera- 
ture of  the  substance,  and  hence 


'-T-*  ••••;  {48) 

since  by  the  help  of  the  Differential  Calculus 

/dp\          fdv\    //dv\  l/5v\  l/dv\ 

(ay.-  -  (*f),/  y  r'  a-  w;,«     ^=  "fe)- 

where  a  denotes  the  coefficient  of  expansion  at  constant 
pressure,  and  0  the  coefficient  of  compression  at  constant 
temperature  of  the  substance.  These  coefficients  can  be 
measured  directly  and  hence  numerical  values  of  Pn  be  ob- 
tained. The  values  obtained  in  this  way  are  likely  to  be 
very  approximately  correct,  since  the  dissociation  of  the 
molecules  with  the  expansion  of  a  substance  is  usually 
negligible,  and  the  change  in  internal  molecular  energy  we 
would  expect  to  be  small  in  comparison  with  the  work 


72         THE  EFFECT  OF  THE  MOLECULAR    FORCES 

done  against  molecular  attraction,  i.e.,  against  the  intrinsic 
pressure. 

Table  III  gives  the  values  of  the  intrinsic  pressure  at 
different   temperatures   for   a   few   liquids   calculated*   by 


TABLE  III 


ETHER,  (C4H10O). 

TCL 

t°c 

010« 

or 

P 

rn=-Q- 

p'atmos. 

atmos. 

13.5 

169 

.001574 

.7214 

2669 

232.3 

25.4 

190 

.001632 

.7077 

2467 

237.3 

63 

300 

.001809 

.6620 

2026 

250.0 

78.5 

367 

.001892 

.6421 

1812 

253.6 

99 

539 

.001992 

.6105 

1375 

255.2 

BENZENE,  (C6H6). 

15.4 

87 

.001215 

.8840 

4083 

271.8 

50.1 

111 

.001305 

.8466 

3796 

291.6 

78.8 

126 

.001379 

.8145 

3850 

305.5 

CHLOROFORM,  (CHC13). 

0 

101 

.001107 

1.5264 

2991 

289.8 

20 

128 

.001294 

1.4885 

2970 

303.7 

40 

162 

.001484 

1.4503 

2869 

315.8 

60 

204 

.001670 

1.4108 

2726 

326.9 

PENTANE,  (C5Hi2). 

0 

229 

.001465 

.6454 

1747 

203.5 

20 

318 

.001589 

.6262 

1337 

211.9 

40 

416 

.001721 

.6062 

1294 

219.2 

60 

486 

.001830 

.5850 

1260 

225.0 

*  R.  D.  Kleeman,  Proc.  Camb.  Phil  Soc.,  Vol.  XVI,  Pt.  6  (1912), 
p.  545. 


THE  INTRINSIC  PRESSURE  EQUATION  73 

means  of  the  foregoing  formula,  and  for  comparison  the 

r>/77 

values  of  the  external  pressures  p'  given  by  p' '  — which 

Tfl 

the  substances  would  have  if  they  behaved  as  perfect  gases. 
The  values  of  the  coefficient  /3  in  the  Table  refer  to  the 
compression  per  atmosphere,  and  hence  the  values  of  Pn  are 
expressed  in  terms  of  the  same  quantity.  The  values  of  p' 
have  therefore  also  been  expressed  in  terms  of  this  quantity 
by  dividing  the  value  of  R  in  the  gas  equation  (Section  5) 
by  106.  The  values  of  a,  /3,  and  p,  were  taken  from  Landolt 
and  Bernstein's  Tables,  the  values  of  p  being  unnecessary, 

since  they  are  so  small  in  comparison  with  those  of  T- 

p 

that  they  may  be  neglected.  The  values  of  Pn  are  striking 
on  account  of  their  magnitude,  being  much  greater  than 
p',  and  illustrate  the  powerful  attraction  that  exists  between 
molecules  for  close  distances  of  approach.  Since  the  attrac- 
tion of  a  molecule  probably  increases  with  its  mass,  the 
magnitude  of  the  intrinsic  pressure  for  constant  volume 
probably  varies  in  a  similar  way.  It  would  evidently  de- 
crease with  the  density  p  of  the  substance  since  the  attrac- 
tion between  two  molecules  decreases  with  their  distance  of 
separation. 

The  intrinsic  pressure  is  evidently  greater  than  the 
external  pressure,  and  this  holds  therefore  also  for  the 
expansion  pressure  Pe  according  to  equation  (45),  or  Pn>p 
and  Pe>p.  Also  according  to  this  equation  Pe>Pn,  and 
since  Pn>p'  according  to  the  Table,  we  also  have  Pe>p'. 
The  result  that  the  expansion  pressure  Pe  of  a  liquid  is 
greater  than  the  external  pressure  p'  it  would  have  if  it 
behaved  as  a  perfect  gas  is  caused  according  to  Section  20 
by  the  molecules  possessing  an  apparent  volume,  and  that 
the  total  average  velocity  of  a  molecule  is  greater  than  the 
velocity  it  would  have  in  the  perfectly  gaseous  state,  or 
that  6 >0  and  Vt>Va. 


74  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

If  the  equation  of  state  (Section  26)  of  a  substance  be 
known  the  values  of  a  and  )8  may  be  calculated  for  different 
temperatures  and  densities  of  a  substance  by  means  of  the 
equations  defining  these  quantities,  or  better,  the  value  of 

( -—=,  )  may  be  directly  calculated  and  hence  the  correspond- 

\0l/t 

ing  intrinsic  pressure  be  obtained. 

It  is  possible  now  to  obtain  superior  and  inferior  limits 
of  the  values  of  n  and  Vt  of  a  substance. 

22.  Superior   and   Inferior    Limits   of  n   of  a 
Substance. 

It  will  appear  from  an  inspection  of  equation  (46)  that 
a  superior  limit  of  n,  the  number  of  molecules  crossing  a 
square  cm.  from  one  side  to  the  other  per  second,  which 
will  be  written  n",  is  obtained  by  supposing  that  the  appar- 
ent volume  of  the  molecules  is  zero,  or  6  =  0,  in  which  case 
the  equation  becomes 


=  2.543X10-^ 

The  quantity  Pn  on  the  right  hand  side  of  this  equation, 
we  have  seen,  can  be  calculated  from  quantities  determined 
by  experiment. 

If  Pn  is  zero,  Vt  has  the  same  value  as  if  the  substance 
were  in  the  perfectly  gaseous  state,  since  it  is  independent 
of  the  apparent  molecular  volume  (Section  19).  Hence 
according  to  equations  (35)  and  (8),  the  value  of  n  under 
these  conditions,  which  will  be  written  n',  is  given  by 


(50, 


This  is  an  inferior  limit  of  n  since  Vt  cannot  have  a  smaller 
value  than  corresponding  to  the  perfectly  gaseous  state, 


THE  INTRINSIC  PRESSURE  EQUATION 


75 


Table  IV  contains  as  an  illustration  the  superior  and 
inferior  limits  of  n  for  a  few  substances  in  the  liquid  state*. 
The  values  of  Pn  used  in  the  calculations  are  given  in  the 
Table  III. 

TABLE  IV 


ETHER 

PENTANE 

*°C. 

Sup.  Lim. 
of  n  ID-2*. 

Inf.  Lim. 
of  n  10-25. 

t°  C. 

Sup.  Lim. 
of  n  ID-2*. 

Inf.  Lim. 
of  n  10-25. 

13.5 

78.5 

71 
44 

6.24 
6.16 

0 
60 

49 

32 

5.23 
5.68 

CHLOROFORM 

BENZENE 

0 
60 

65 
54 

6.30 
6.50 

15.4 

78.8 

108 
94 

7.0 
7.12 

It  will  be  seen  that  in  the  case  of  each  substance  one  of 
the  limits  is  many  times  that  of  the  other,  the  actual  value 
of  n  of  course  lying  between  the  two.  Since  the  values  of 
Pn  are  large  in  comparison  with  the  values  of  p'  (the  external 
pressure  a  substance  would  have  in  the  perfectly  gaseous 
state  (Section  21)),  and  the  former  are  the  outcome  of  molec- 
ular attraction  which  in  proportion  effects  the  velocities 
of  translation  of  the  molecules,  and  therefore  the  values  of 
n,  we  would  expect  the  actual  values  of  n  to  lie  nearer  to 
the  superior  than  to  the  inferior  limits  obtained. 


23.  Superior  and  Inferior  Limits    of  Vt  of  a 
Substance. 

These  limits  are  obtained    by  substituting  in  equation 
(35)  the  superior  and  inferior  limits  of  n  obtained  by  the 

*  If  the  values  of  n  are  corrected  according  to  Maxwell's  law  they 
have  to  be  divided  by  1.085  according  to  Section  20. 


76 


THE  EFFECT  OF  THE  MOLECULAR  FORCES 


method  given  in  the  previous  Section.  Table  V  gives  the 
values  obtained  in  this  way  for  a  few  substances.  For 
reasons  stated  in  the  previous  Section  we  would  expect 
that  the  actual  total  average  velocity  of  a  molecule  in  a 
substance  sliould  lie  nearer  '/o  the  superior  than  to  the  inferior 
limit.  The  inferior  limit  is  the  velocity  when  the  substance 
is  in  the  perfectly  gaseous  state. 


TABLE  V 


ETHER 

PENTANE 

t°C. 

Sup.  Lim. 

w^: 

Inf.  Lim. 

-  v>w'™: 

t°c. 

Sup.  Lim. 

of  F,  10*  5™' 

1         sec. 

Inf.  Lim. 

of  F  104^ 
1         sec. 

13.5 

35.7 

3.10 

0 

26.4 

2.83 

CHLOROFORM 

.  BENZENE 

0 

24.6 

2.38 

15.4 

46.0 

2.98 

The  quantities  Vt  and  n  do  not  depend  on  the  quantity 
b  by  definition  according  to  Section  19.  Therefore  if  Pn 
and  p  were  also  independent  of  6,  equation  (49)  would  give 
the  actual  value  of  n  instead  of  a  superior  limit,  while  equa- 
tion (35)  would  give  the  actual  value  of  Vt.  But  p  obviously 
depends  on  b  (Section  19) ;  it  is,  however,  small  in  comparison 
with  Pn  in  the  cases  considered.  Therefore  if  b  did  not  depend 
on  the  molecular  forces  Pn  would  be  independent  of  6, 
and  the  superior  limits  of  n  and  Vt  obtained  would  very 
approximately  represent  the  actual  values  of  these  quan- 
tities. There  is  some  indirect  evidence,  however,  with 
which  we  will  get  acquainted  as  we  proceed,  that  b  is  the 
outcome  of  molecular  forces,  as  we  would  expect. 

The  mathematical  definition  of  b  was  given  in  Section 
19,  but  so  far  no  information  about  its  value  and  properties 


THE  CAUSE  OF  INTERNAL  MOLECULAR  VOLUME      77 

has  been  discussed.     It  will  be  convenient  now  to  discuss 
the  quantity  from  a  certain  aspect. 

24.  The    Real    and    Apparent    Volumes    of    a 
Molecule,  and  their  Superior  Limits. 

The  property  of  volume  of  a  substance  is  known  to  us 
only 'through  the  resistance  to  force:  thus  if  two  portions 
of  matter  are  pressed  together  there  is  a  resistance  to  the 
operation  which  need  not  be  accounted  for  by  actual  con- 
tact taking  place — whatever  that  may  mean,  but  by  an 
approach  of  the  molecules  of  the  substances  to  such  dis- 
tances that  the  resultant  of  their  forces  of  repulsion  equals 
the  force  with  which  the  substances  are  pressed  together. 
It  seems  unsafe  therefore,  and  little  warranted  by  the  facts, 
to  associate  with  an  atom  an  impenetrable  perfectly 
elastic  volume  of  constant  magnitude.  There  is  no  doubt, 
though,  that  it  may  be  said  that  a  volume  is  associated 
with  each  molecule  through  which  the  center  of  another 
molecule  cannot  pass,  but  whose  magnitude  depends  upon 
external  conditions.  If  this  volume  is  due  to  the  existence 
of  forces  of  repulsion,  as  is  highly  probable,  its  magnitude 
will  depend  upon  the  force  exerted  in  approaching  another 
molecule.  Thus  it  is  evident  that  in  the  case  of  a  gas  this 
volume  will  decrease  with  increase  of  temperature,  for  this 
is  attended  by  an  increase  in  the  kinetic  energy  of  the  mole- 
cules, and  therefore  by  a  decrease  in  the  minimum  distance 
of  approach,  the  molecules  approaching  each  other  to  a 
distance  at  which  their  kinetic  energy  is  completely  or 
partly  converted  into  potential  energy  of  repulsion.  It 
seems  futile,  therefore,  to  expect  to  obtain  a  constant  value 
for  the  quantity  in  question  since  it  varies  with  the  external 
conditions,  which  often  are  not  altogether  known.  The 
best  we  can  hope  to  obtain  is  the  order  of  magnitude  of  the 
quantity,  which  is  likely  to  be  always  the  same. 


78          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

The  apparent  volume  of  a  molecule,  or  of  a  gram  mole- 
cule of  molecules,  which  was  investigated  in  Section  19, 
is,  however,  a  perfectly  definite  mathematical  quantity. 
But  its  connection  with  the  "real  volume,"  whatever  that 
may  mean,  cannot  be  ascertained  with  certainty,  if  at  all, 
since  the  apparent  volume  would  depend  on  the  geometrical 
configuration  of  the  "real  volume."  Thus  for  example, 
Van  der  Waals  has  shown  that  if  the  "real  volume"  of  a 
molecule  consists  of  an  impenetrable  sphere,  the  apparent 
volume  is  four  times  that  of  the  "real  volume."  Now  if 
the  apparent  volume  is  caused  by  forces  of  repulsion  between 
the  molecules,  it  is  difficult,  if  not  impossible,  to  define  exactly 
what  is  meant  by  the  "real  volume,"  and  therefore  to  deter- 
mine its  magnitude.  It  will  be  evident  on  reflection  that 
the  "real  volume"  of  a  molecule  is  not  exactly  the  volume 
through  which  another  molecule  will  never  pass. 

The  volume  occupied  by  a  molecule,  or  associated  with 
a  molecule,  at  absolute  zero,  is  of  special  interest  in  this 
connection.  Since  at  that  temperature  the  molecules 
have  no  motion  of  translation  each  will  take  up  a  position 
so  that  the  forces  of  attraction  and  repulsion  acting  upon 
it  balance  each  other,  in  other  words,  since  Pe  and  p  are 
zero  Pn  is  zero  according  to  equation  (45).  The  apparent 
and  "real  volume"  are  evidently  equal  to  each  other  under 
these  conditions.  The  apparent  volume  of  a  molecule  at 
the  absolute  zero  is  likely  to  be  greater  than  at  a  higher 
temperature,  since  the  velocity  of  the  molecules  due  to 
their  temperature  is  likely  to  make  the  molecules  approach 
closer  to  each  other  in  opposition  to  their  repulsion  (Sec- 
tion 15)  than  they  would  otherwise.  This  volume  is  thus 
a  superior  limit  of  the  apparent  molecular  volume  for  higher 
temperatures  than  the  absolute  zero.  An  approximate 
value  of  this  volume  at  the  absolute  zero  may  be  obtained 
from  considerations  involving  the  coefficient  of  expansion, 
or  by  means  of  Cailletet  and  Mathias'  linear  diameter 


THE  MOLECULAR  VOLUME  AT  THE  ABS.  ZERO       79 


law.  According  to  this  law  the  densities  pi  and  p2  of  a  liquid 
and  its  saturated  vapor  respectively  are  given  by  the  equa- 
tion 


where  T  denotes  the  absolute  temperature,  and  a  and  c 
denote  constants  depending  only  on  the  nature  of  the 
liquid.  These  constants  may  be  determined  from  values 
of  pi  and  P2  corresponding  to  two  different  temperatures 
of  the  liquid.  At  the  absolute  zero,  or  T  =  0,  we  accordingly 
have  pi  =  a,  since  p2  =  0  and  77  =  0.  As  an  illustration  the 
values  of  v0  of  a  gram  molecule  for  a  number  of  substances 
at  the  absolute  zero,  or  the  values  of  ra/pi,  and  the  cor- 
responding values  of  V'Q  and  (Vo)^  for  a  single  molecule, 
are  given  in  Table  VI.  It  will  be  seen  that  the  values  of 

TABLE  VI 


Substance. 

V 
c.c. 

V 
c.c. 

i 

vo 

V  io«. 

c.c. 

cv>x 

108. 

cm. 

Oxygen 

20.8 

49.2 

2.37 

33.5 

3.22 

Nitrogen  

25.0 

70 

2.80 

40.2 

3.42 

Carbon  dioxide  
Ether                      .... 

25.5 
71.7 

96 
280 

3.77 
3.91 

41.1 
115 

3.44 

4.87 

Benzene  
Carbon  tetrachloride  .  . 
Propyl  acetate     .... 

70.6 
72.2 

86.2 

256 
276 
345 

3.63 

3.82 
4.00 

114 

116 
139 

4.84 

4.88 
5.18 

V'Q  increase  with  increase  of  molecular  weight,  as  we  would 
expect.  The  values  of  (i/o)**  probably  give  the  order  of  mag- 
nitude of  the  apparent  diameter  of  a  molecule  under  various 
conditions.  The  Table  also  gives  the  values  of  the  critical 
volume  ve,  and  the  values  of  the  ratio  VC/VQ,  which,  it  will 
be  seen,  lie  in  the  neighborhood  of  4.  The  values  of  VQ 
and  vc  were  taken  from  a  Table  given  in  Nernst's  Theo- 
retische  Chemie,  7th  ed.,  p.  234.  A  superior  limit  of  b  is 
thus  vc/4. 


80 


THE  EFFECT  OF  THE  MOLECULAR  FORCES 


An  important  application  of  the  foregoing  results  may 
immediately  be  made. 


25.  Inferior  Limits  of  n  and  Vt. 

The  results  of  the  preceding  Section  enable  us  to  obtain 
another  inferior  limit  for  n  of  a  substance  by  means  of 
equation  (46),  on  writing  b  =  vc/4i,  which  is  a  superior  limit 
of  6.  On  substituting  the  inferior  limit  of  n  thus  obtained 
in  equation  (35)  it  gives  an  inferior  limit  of  Vt.  Table  VII 

TABLE  VII 


cms. 
CO2  AT  40°  C.     Fa  =  4.165Xl04  - 

Sec. 

p  in 
atmos. 

Inf.  Lim. 
of  n  1Q26. 

Inf.  Lim. 

of  vt  io<  ?5»- 
1         sec. 

V  in 
atmos. 

Inf.  Lim. 
of  n  10». 

Inf.  Lim. 

of  Vt  10«  C*L 
sec. 

70 

3.88 

4.60 

94 

14.4 

5.94 

80 

5.76 

4.79 

100 

16.4 

6.24 

85 

8.01 

5.06 

112 

18.7 

6.63 

contains  the  inferior  limits  for  C02  at  40°  C.  calculated  from 
the  values  of  Pn+p  obtained  in  Section  34  and  given  in 
Table  XIV.  It  will  be  seen  that  at  constant  temperature 
the  inferior  limit  of  Vt,  the  total  average  molecular  velocity, 
gradually  increases  with  increase  of  pressure  or  density  of 
the  substance,  and  that  its  value  corresponding  to  an  external 
pressure  of  112  atmos.  is  about  50  per  cent  greater  than  the 
values  of  Va,  the  total  average  velocity  of  a  molecule  of  the 
substance  in  the  perfectly  gaseous  state  at  the  same  tempera- 
ture*. This  definitely  shows  that  the  total  average  molec- 

*  The  values  obtained  for  Vt  and  Va  have  not  been  corrected  for  an 
uneven  distribution  of  molecular  velocities.  According  to  Maxwell's 
Law  of  Distribution  each  value  has  to  be  divided  by  1.085  according  to 
Sections  8,  11,  and  20. 


DENSITY  AND  MOLECULAR  VELOCITY  81 

ular  velocity  at  constant  temperature  in  a  substance  not 
obeying  the  laws  of  a  perfect  gas  increases  with  the  density, 
and  that  it  may  be  considerably  larger  than  the  velocity 
when  the  substance  is  in  the  perfectly  gaseous  state.  The 
results  thus  fall  into  line  with  those  of  Sections  16  and  17, 
and  we  will  see  later  with  those  of  Section  29. 

From  this  it  follows  that  n  according  to  equation  (35) 
increases  more  rapidly  with  the  density  of  a  substance 
than  proportionally  to  it. 

In  the  previous  Sections  we  have  mainly  considered 
individual  properties  of  molecules.  It  will  be  of  interest 
and  importance  now  to  consider  some  properties  of  a  sub- 
stance as  a  whole  which  depend  on  the  dynamical  and 
attractional  properties  of  the  molecules. 

26.  The  Equation  of  State  of  a  Substance. 

The  state  of  a  given  mass  of  a  substance  according  to 
experiment  is  completely  defined  by  two  variables.  There- 
fore each  of  the  variables  of  a  substance  may  be  expressed 
in  terms  of  any  two  by  means  of  an  equation.  Such  a  rela- 
tion is  called  an  equation  of  state  of  the  substance,  and 
contains  two  independent  variables.  The  most  useful 
equation  is  that  connecting  the  variables  pressure,  volume, 
and  temperature. 

One  of  the  equations  of  state  is 

Pe,      ......     (51) 


obtained  in  Section  21,  where  p,  Pn,  and  Pe  denote  the 
external,  intrinsic,  and  expansion  pressures  respectfully 
of  the  substance.  This  equation  may  also  be  written 


=  2.534  X10-20\7X  .     .     .     (52) 

according  to  equation  (38),  where  n  denotes  the  number  of 
molecules  crossing  a  square  cm.  per  second,  and  6  the  appar- 


82          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

ent  volume  of  the  molecules.  According  to  equation  (40) 
and  subsequent  remarks  the  preceding  equation  may  also 
be  written 


(53) 


where  Vt  denotes  the  total  average  velocity  of  a  molecule 
of  a  substance  for  any  state,  Va  the  average  velocity  in  the 
gaseous  state,  v  the  volume  of  a  gram  molecule,  6  the  appar- 
ent molecular  volume,  and  R  the  gas  constant.  The  fore- 
going equations  are  fundamental  in  character,  since  they  do 
not  depend  upon  any  hypothesis. 

If  equation  (52),  (like  (53)),  is  corrected  according  to 
Maxwell's  law  of  distribution  of  molecular  velocities  in  the 
gaseous  state,  the  right-hand  side  of  the  equation  has  to 


be  multiplied  by  *-«•,  or  1.085,  according  to  Section  20. 

The  problem  that  remains  to  be  solved  in  connection 
with  the  foregoing  equations  is  to  find  an  expression  for  the 
quantities,  Pe,  n,  Vt,  and  6,  in  terms  of  other  quantities, 
particularly  in  terms  of  the  quantities  p,  v,  and  T,  which 
can  be  measured  directly.  This  cannot  be  done  without 
the  introduction  of  assumptions,  and  therefore  yields  results 
which  are  likely  to  hold  only  approximately. 

Van  der  Waals  has  proposed  the  equation  of  state 

t  &      RT  /*.  *\ 

p+-2  =  —  ......     (54) 


where  a  and  b  are  constants.  On  comparing  this  equation 
with  equation  (53)  we  see  that  Pn  =  a/v2,  and  Vt=Va. 
According  to  the  latter  equation  the  velocity  of  translation 
of  the  molecules  is  not  influenced  by  molecular  attraction. 
But  this  is  inadmissible  according  to  Sections  17,  20,  and 
25,  nor  does  it  hold  approximately,  for  we  will  see  in  Section 


VAN  DER  WAALS'  EQUATION  OF  STATE  83 

29  that  when  a  substance  is  in  the  liquid  state  the  value  of 
Vt  may  be  more  than  four  times  the  value  of  Va.  Thus 
the  values  of  b  calculated  from  Van  der  Waals'  equation 
would  be  too  large. 

Van  der  Waals  deduced  his  equation  from  the  supposi- 
tions that  the  total  average  velocity  of  a  molecule  is  not 
affected  by  molecular  attraction,  or  is  the  same  as  the 
velocity  in  the  gaseous  state,  and  b  represents  the  apparent 
volume  of  a  gram  molecule  of  molecules  as  a  whole  due  to 
a  true  impenetrable  spherical  volume  being  associated 
with  each  molecule.  The  value  of  b  is  then  four  times  the 
true  volume  occupied  by  the  molecules. 

The  equation  from  Van  der  Waals  as  a  whole,  however, 
fairly  well  represents  the  facts,  that  is,  the  values  of  each  of 
the  quantities  p,  v,  and  T,  may  be  obtained  by  means  of 
it  given  the  corresponding  values  of  the  remaining  two 
quantities.  The  equation  is  particularly  useful  for  such 
determinations  on  account  of  its  simplicity. 

It  can  easily  be  shown  that  the  intrinsic  pressure  term 
a/v2  in  Van  der  Waals'  equation  corresponds  to  attraction 
between  two  molecules  varying  inversely  as  the  fourth  power 
of  their  distance  of  separation.  Thus  the  molecules  of  a 
substance  may  be  divided  into  a  number  of  parallel  rows 
separated  from  each  other  by  a  distance  xi,  the  distance 
of  separation  of  the  molecules.  Therefore  if  A\/xs  repre- 
sents the  law  of  molecular  attraction,  the  attraction  of  one- 
half  of  a  row  on  the  other  half  may  evidently  be  written 
A2/xs,  where  A 2  represents  an  appropriate  constant.  Since 
l/x2  rows  pass  through  a  square  cm.  situated  at  right  angles 
to  the  rows,  the  attraction  of  the  molecules  in  one-half  of  a 
cylinder  of  unit  cross-section  and  infinite  length  on  the 
molecules  in  the  other  half  is  given  by 


A2 

xs+2> 


84          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

and  this  is  the  intrinsic  pressure.    Now  x\  —  ( — )    =  ( — - )   , 

\  P/        \  wi  / 

where  p  denotes  the  density,  and  v  the  volume  of  a  gram 
molecule  of  the  substance,  and  thus  the  intrinsic  pressure 
may  be  written 


4 

A3p  3  ,    or    - 


s-f-2 
where  A  3  and  A  4  are  constants.    Therefore  when  -—  =  2, 

o 

s  =  4. 

The  effect  of  molecular  attraction  is,  however,  not  so 
simple  that  it  may  be  expressed  by  a  single  term.  According 
to  Section  14  it  should  contain  at  least  two  additional  terms 
having  negative  signs  which  express  repulsion  between  the 
molecules.  The  attraction  terms  are  probably  approxi- 
mately represented  by  the  foregoing  single  term  involving 
the  inverse  fourth  power.  The  subject  of  molecular  attrac- 
tion and  its  bearing  on  the  form  of  the  intrinsic  pressure 
term  is  not  of  primary  importance  in  connection  with  the 
object  of  this  book,  and  will  therefore  not  be  discussed 
here  any  further  but  reserved  for  another  place.  For  pur- 
poses of  calculating  the  values  of  one  of  the  variables  p,  v, 
and  T,  from  given  values  of  the  other  two  variables  most 
of  the  empirical  equations  of  state  answer  equally  well. 
An  account  of  a  number  of  them  will  be  found  in  Winckel- 
mann's  Handbuch  der  Physik,  IT.  Edition,  pp.  1135-1143. 

Equation  (53)  may  be  given  a  form  by  means  of  which 
the  values  of  Vt  and  Pn  of  a  substance  may  approximately 
be  calculated.  The  quantity  Vt  may  be  taken  equal  to 
the  sum  of  two  terms,  one  equal  to  Va,  and  the  other  equal 
to  Vi,  which  represents  the  effect  of  molecular  attraction 
on  the  value  of  Vt.  Now  a  molecule  will  pass  over  a  distance 
propprtjonal  to  x\  in  passing  out  of  the  influence  of  one  mole- 


A  USEFUL  FORM  OF  THE  EQUATION  OF  STATE      85 

cule  and  under  the  influence  of  another.  Therefore  if  A\/xs 
represents  the  law  of  molecular  attraction  the  change  of 
kinetic  energy  of  the  molecule  over  that  distance  may  be 

written  ~^1-,  and  the  velocity  V\  is  therefore  approximately 

given  by 

D2  A? 


where  D$  is  a  constant.     Equation  (53)  may  accordingly 
be  written 

A*  Z)3       RT  . 

'      '    '    '(55) 


o 

where    Va=  \l7r\ according   to    Sections   6   and   8. 

\OTT\    m 

On  applying  this  equation  to  three  states  of  a  substance  not 
differing  much  from  one  another  at  constant  temperature 
the  quantities  A*,  Ds,  and  b,  may  be  taken  as  constant  and 
determined  from  these  equations.  The  values  of  Vt  and 
Pn  corresponding  to  these  three  states  may  then  at  once  be 
calculated.  Similarly  the  equation  may  be  applied  to  another 
set  of  three  states  and  the  corresponding  values  of  Vt,  Pn 
and  b,  obtained,  and  so  on.  The  values  of  Vt  and  b  obtained 
in  this  way  by  means  of  equation  (55)  will  obviously  not 
be  so  accurate  as  those  obtained  by  means  of  equation  (67) 
in  Section  29.  The  former  equation  can,  however,  be  more 
readily  applied  to  the  facts.  The  values  of  6  obtained  by 
this  equation  should  more  accurately  represent  the  facts 
according  to  the  definition  of  b  in  Section  19  than  those 
obtained  by  Van  der  Waals'  equation,  since  Vt  is  not  taken 
equal  to  Va.  If  s  is  not  known  it  may  be  determined  by 
applying  equation  (55)  to  four  different  states  of  the  sub- 
stance. 


86          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

The  quantities  Pe  and  Pn  in  the  equation  of  state  must 
very  approximately  satisfy  the  relation  obtained  on  eliminat- 
ing p  from  equations  (51)  and  (48),  which  gives 


It  follows  from  this  equation  that  if  Pn  is  a  function  of  v 
only  Pe  is  of  the  form  T-4>(y),  where  (j>(v)  is  a  function  of  v. 
Van  der  Waals'  equation  of  state  satisfies  the  above  condi- 
tion, but  of  course  it  does  not  follow  that  therefore  its  form 
is  correct.  It  is  fairly  certain  that  Pn  is  a  function  of  T 
as  well  as  of  v.  For  the  intrinsic  pressure  of  a  substance, 
since  it  depends  upon  molecular  attraction,  will  depend  upon 
the  change  in  atomic  configuration  of  a  molecule  with  change 
in  temperature.  Since  such  a  change  undoubtedly  occurs, 
as  is  shown  by  the  deviation  of  the  ratio  of  the  specific  heats 
considered  in  Section  13  from  the  value  1.66,  the  form  of 
the  function  for  Pe  is  not  as  simple  as  a  linear  function  of  T. 

27.  The  Conditions  that  the  Equation  of  State 
has  to  satisfy. 

If  the  pressure  of  a  substance  is  plotted  against  its  volume 
at  different  temperatures,  curves  of  the  form  shown  in  Fig. 
8  are  obtained.  For  certain  temperatures  each  curve  con- 
sists of  two  parts,  each  of  which  has  one  of  the  terminal 
points  lying  on  a  line  (dotted  in  the  figure)  parallel  to  the 
volume  axis.  Thus  corresponding  to  the  pressure  indicated 
by  the  dotted  line  the  substance  can  only  have  either  of 
the  two  volumes  corresponding  to  the  abscissae  of  the 
points  at  the  extremities  of  the  line.  As  the  temperature 
of  the  substance  is  increased  the  length  of  the  dotted  line 
is  decreased,  and  for  a  certain  temperature,  called  the  critical 
temperature,  it  entirely  disappears. 


THE  CONTINUITY  OF  STATE 


87 


Van  der  Waals  has  proposed  a  theory  to  explain  this 
behavior  of  substances.  According  to  it  the  two  curves 
corresponding  to  a  given  temperature  are  theoretically 
parts  of  a  single  curve  as  shown  in  Fig.  9,  the  form  of  the 
absent  part  being  such  as  to  render  the  curve  continuous. 
The  fact  that  the  intermediate  part  of  the  curve  is  not 
realized  in  practice  is  explained  by  supposing  that  a  sub- 


FIG.  8. 


stance  in  each  of  the  states  represented  by  it  is  in  unstable 
equilibrium,  accordingly  a  disturbance  from  the  outside 
would  make  the  substance  change  to  one  or  both  the  stable 
forms  represented  by  the  points  a  and  6  at  the  extremities 
of  this  part  of  the  curve. 

Thus  according  to  this  theory  there  is  no  discontinuity 
of  state;    and  therefore  the  equation  connecting  p  and  v 


88 


THE  EFFECT  OF  THE  MOLECULAR  FORCES 


for  the  two  stable  portions  of  a  curve  will  also  represent  the 
unstable  condition. 

It  will  appear  from  an  inspection  of  the  curves  in  Fig.  9 
that  the  point  e,  which  corresponds  to  the  critical  point, 


FIG.  9. 

is  a  point  of  inflexion.     The  geometrical  properties  of  the 
point  are  expressed  by  the  equations 


and 


ft]  =0- 

\  oV  /  T 


to2 


(57) 
(58) 


which  the  equation  of  state  has  to  satisfy  at  the  critical 
point.  These  equations  express  relations  between  the  con- 
stants of  the  equations  of  state  and  the  critical  constants. 


CONDITIONS  INVOLVING  THE  ABSOLUTE  ZERO      89 

The  foregoing  equations,  it  may  be  mentioned,  can  also  be 
deduced  from  the  Laws  of  Thermodynamics. 

The  equation  of  state  must  also  possess  the  property 
that  it  vanish  if  the  substitutions  p  =  0,  T  =  0,  and  #  =  <», 
are  made,  which  correspond  to  the  absolute  zero  of  tem- 
perature. This  may  indicate  that  a  relation  exists  between 
the  coefficients  of  the  equation,  or  that  it  has  a  certain  form. 

If  the  temperature  and  volume  at  constant  pressure  of 
a  substance  are  plotted  against  each  other  instead  of  the 
pressure  and  volume  at  constant  temperature,  a  set  of 
curves  similar  to  those  shown  in  Fig.  8  is  obtained.  A 
portion  of  each  curve  cannot  be  realized  in  practice.  The 
theoretical  curve  representing  this  portion  must  for  reasons 
of  continuity  cut  a  volume  ordinate  in  three  points,  similarly 
as  shown  in  Fig.  9.  But  this  cannot  hold  at  the  absolute 
zero,  since  the  absolute  temperature  cannot  have  negative 
values,  and  therefore  no  part  of  the  curve  can  lie  on  the 
negative  side  of  the  volume  axis.  The  part  of  the  curve 
lying  between  the  volumes  VQ  and  infinity  inclusive  in  that 
case  coincides  with  the  volume  axis.  This  condition  is 
expressed  by  the  equation 


which  holds  for  v  =  infinity,  v  —  v0,  and  inclusive  values, 
corresponding  to  p  =  0,  and  T  =  Q.  This  probably  indicates 
that  the  equation  of  state  must  have  a  certain  general  form 
which  satisfies  this  condition. 

Experiment  shows  that  the  variables  pressure,  volume, 
and  temperature,  of  a  liquid  in  contact  with  its  saturated 
vapor,  may  each  be  expressed  by  an  equation  in  terms  of 
one  of  the  variables,  and  this  holds  also  for  the  saturated 
vapor.  It  can  be  shown  that  this  is  a  consequence  of  Van 
der  Waals'  theory  of  the  continuity  of  state,  and  the  Laws  of 
Thermodynamics.  Thus  it  follows  from  thermodynamics 


90  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

that  the  work  done  during  an  isothermal  process  between 
given  limits  is  independent  of  the  path  of  the  process.  Hence 
the  work  done  in  passing  from  the  point  a  to  the  point  b 
in  Fig.  9  is  the  same  for  the  two  paths  indicated,  one  of  which 
corresponds  to  that  obtained  in  practice  with  a  substance, 
while  the  other  to  that  given  by  the  equation  of  state.  This 
condition  is  expressed  by  the  equation 


-t>i)  =   I    p-dv,        ....     (60) 

where  the  suffixes  1  and  2  refer  to  the  liquid  and  vaporous 
states  respectively. 

On  eliminating  two  of  the  variables  pi,  vi,  V2,  and  T 
from  the  foregoing  equation  and  the  two  equations  obtained 
by  applying  the  equation  of  state  to  the  liquid  and  the 
vaporous  state  of  the  substance,  a  number  of  equations  are 
obtained  each  of  which  contains  two  variables  only. 

Another  set  of  equations  each  of  which  contains  two  varia- 
bles only  may  be  obtained  as  follows.*  According  to  Clapey- 
ron's  equation  the  internal  heat  of  evaporation  L  of  a  gram 
molecule  of  a  substance  is  given  by 

L=\T-j^-p\(v2-vi) (61) 

Now  according  to  equation  (47) 


and  therefore  Clapeyron's  equation  by  the    help  of  equa- 
tion (60)  may  be  written 


dv  =  -j7j1(v2  —  vi) (62) 

*  R.  D.  Kleeman,  Phil  Mag.,  Sept.,  1912,  pp.  391-402. 


CONDITIONS  INVOLVING  THERMODYNAMICS        91 

From  this  equation  and  the  equation  of  state,  a  number 
of  equations  may  be  obtained  similarly,  as  before,  each  of 
which  contains  two  variables  only. 

Now  if  one  of  the  variables  is  eliminated  from  two 
equations  taken  from  the  foregoing  two  sets  of  equations 
which  contain  the  same  two  variables,  an  equation  is  ob- 
tained containing  one  variable  only.  This  equation  must 
evidently  identically  vanish  with  respect  to  this  variable. 
This  requires  that  some  of  the  constants  in  the  equation 
be  equated  to  zero.  Since  these  constants  are  functions 
of  the  constants  of  the  equation  of  state,  the  equations  thus 
obtained  express  relations  between  the  latter  constants. 
The  number  of  these  equations  evidently  depends  on  the 
form  assumed  for  the  equation  of  state. 

It  appears,  therefore,  that  the  equation  of  state  has  to 
satisfy  a  number  of  conditions  which  govern  its  form  and 
the  relation  between  its  constants.  Other  equations  of 
condition  besides  those  pointed  out  may  of  course  exist, 
which  have  not  yet  been  discovered. 

The  equation  of  state  possesses  besides  an  important 
property  which  indicates  certain  properties  of  its  constants. 
This  property,  which  is  discussed  in  the  next  Section,  is 
based  upon  the  facts,  but  a  theoretical  reason  may  also  be 
found  for  it. 


28.  The  Relation  of  Corresponding  States. 

The  quantities  p,  v,  and  T,  of  a  substance  may  be  expressed 
as  fractions  or  as  multiples  n,  r^  and  rs,  of  their  critical 
values  thus:  p  =  ripc,  v  =  r2Vc,  and  T  =  r%Tc.  According  to 
the  relation  of  corresponding  states  if  the  value  of  each  of 
two  of  the  quantities  n,  r<i,  and  rs,  for  each  of  a  number 
of  substances  be  taken  the  same,  the  remaining  quantity 
has  the  same  value  for  each  substance.  In  the  case  that 


92          THE  EFFECT  OF  THE  MOLECULAR  FORCES 

each  substance  consists  of  two  phases  in  equilibrium,  and 
one  of  the  foregoing  quantities  is  taken  the  same  for  each 
substance,  each  of  the  remaining  two  quantities  is  the 
same  for  each  substance.  Substances  under  these  condi- 
tions are  said  to  be  at  corresponding  states,  or  obey  the 
relation  of  corresponding  states.  This  relation  was  first 
deduced  by  Van  der  Waals  from  his  equation  of  state  by 
applying  to  it  the  conditions  of  the  critical  point  expressed 
by  equations  (57)  and  (58).  The  resultant  equations 
(one  of  which  is  the  equation  of  state),  on  eliminating  the 
constants  they  contain,  give  an  equation  of  the  form 


which  expresses  the  first  part  of  the  relation  in  question. 
The  second  part  follows  from  the  fact  that  the  equations 
of  state  of  each  of  the  phases  of  the  two  phases  in  equilib- 
rium can  be  obtained  from  the  foregoing  equation  and  the 
thermodynamical  equation  (60),  giving  three  equations 
each  of  which  involves  only  two  of  the  above  ratios. 

Meslin*  has  investigated  the  general  conditions  under 
which  the  relation  may  be  deduced.  The  equation  of  state 
in  its  most  general  form  is 

lMP»  v,  T,  0i,  02,  •  .  .  )=0, 

where  gi,  g%  .  .  .  ,  are  constants  depending  only  on  the 
nature  of  the  substance.  Suppose  that  the  various  equations 
of  condition  applying  to  the  equation  of  state  are  equal 
in  number  to  the  foregoing  constants.  On  eliminating 
these  constants  an  equation  is  obtained  which  is  of  the  form 


v,  T,pc,vc,  Tc)  =  0. 
*  Comp.Jlend.,  116,  135,  1893. 


CORRESPONDING  STATE  CONDITION  EQUATIONS         93 

This  equation  must  be  independent  of  the  nature  of  the 
units  chosen  and  therefore  be  of  the  form 


Therefore  if  fa,  (or  ^i),  has  the  same  fundamental  form  for 
each  substance  the  relation  of  corresponding  states  at  once 
follows. 

The  number  of  equations  of  condition  applying  to  a 
substance  is  not  exactly  known  according  to  the  previous 
Section,  but  it  is  evidently  greater  than  two,  and  the  num- 
ber of  constants  that  must  be  associated  with  an  equation 
of  state  to  obtain  the  relation  of  corresponding  states  is 
therefore  at  present  undetermined.  If  we  are  given  the 
relation  of  corresponding  states,  it  obviously  follows  that 
the  number  of  the  foregoing  equations  of  condition  is  equal 
to  the  number  of  constants  in  the  equation  of  state. 

The  relation  has  been  tested  by  a  number  of  observers 
of  whom  Young  may  be  especially  mentioned,  and  found 
to  agree  fairly  well  with  the  facts.  (An  account  of  these 
investigations  will  be  found  in  Winckelmann's  Handbuch 
der  Physik,  2d  Edition,  pp.  936-946).  The  deviations  that 
occur  are  probably  entirely  due  to  polymerization,  which 
renders  a  substance  a  mixture,  and  to  which  the  relation- 
ship is  not  likely  to  apply. 

The  relation  may  be  extended  to  a  number  of  other 
quantities  besides  those  mentioned  which  occur  frequently 
in  this  book.  According  to  Section  21 


On  substituting  for  p,  v,  and  T  from  the  equations  expressing 
these  quantities  in  terms  of  their  critical  values  we  obtain 

r**  •  •  •  (63) 


94  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

where  r±  is  a  corresponding  state  quantity.    At  the  critical 
point  the  equation  becomes 


where  r'$  is  a  constant,  and  thus  the  preceding  equation 
may  be  written 

Pn  =  r5Pnc,      ......     (64) 

where  r?>  is  a  factor  obeying  the  relation  of  corresponding 
states,  and  Pnc  denotes  the  intrinsic  pressure  at  the  critical 
point.  Equation  (63)  can  immediately  be  tested  by  the 
facts.  The  intrinsic  pressures  of  ether  and  benzene  in  the 
liquid  states  at  temperatures  corresponding  to  277c/3  can 
be  calculated  by  the  method  of  Section  21  and  are  found 
to  be  2756,  and  3893  atmos.  respectively.  The  ratio  of 
these  pressures  is  .708,  while  the  corresponding  ratio  of 
the  critical  pressures  is  .733,  which  is  practically  equal  to 
the  preceding  ratio,  as  should  be  the  case. 

It  can  be  shown  in  a  similar  way  that  the  quantities 
n"  and  V"t,  which  are  superior  limits  of  the  quantities  n 
and  Vt,  Sections  (22)  and  (23),  should  also  obey  the  rela- 
tion of  corresponding  states. 

If  the  quantity  b  in  the  equation  (52)  may  be  expressed 
as  a  fraction  obeying  the  relation  of  corresponding  states 
of  the  critical  volume  vc,  or  of  the  volume  at  the  absolute 
zero,  which  is  highly  probable,  it  can  be  deduced  in  a  similar 
way  from  this  equation  and  equations  (63)  and  (35)  that 
n  and  Vt  obey  the  relation  of  corresponding  states.  But 
even  if  b  did  not  possess  this  property,  Vt  would  obey  the 
foregoing  relation  approximately  since  b  is  usually  small 
in  comparison  with  v. 

It  will  be  of  importance  now  to  discuss  a  method  of  ob- 
taining the  actual  values  of  the  foregoing  quantities  in  any 
given  case. 


AN  EXPRESSION  FOR  THE  QUANTITY  n  95 

29.  The    Determination    of    the    Quantities    n, 
Vt,  and  b,  of  a  Substance. 

These  quantities  may  most  conveniently  be  determined 
by  means  of  the  equation 


20v'  •  •  (65) 


obtained  from  equations  (46)  and  (48)  in  Section  21.  We 
have  seen  that  the  quantity  6  is  likely  to  vary  little  with 
the  density  of  a  substance.  Therefore  over  small  regions 
of  densities  it  may  be  taken  as  constant.  The  quantity 
n  of  a  substance  we  would  expect  to  increase  more  rapidly 
than  proportional  to  the  density.  If  the  substance  behaved 
as  a  perfect  gas  n  would  be  proportional  to  the  density,  or 
inversely  proportional  to  the  volume  according  to  Section 
11.  For  densities  at  which  the  substance  does  not  obey 
the  law  of  a  perfect  gas  the  molecular  attraction  according 
to  Section  25,  has  the  effect  of  increasing  n  above  the  value 
corresponding  to  the  density.  We  may  therefore  suppose 
n  to  consist  of  the  sum  of  a  number  of  terms,  the  first  expres- 
sing the  effect  of  the  average  minimum  velocity  (Section  16) 
on  the  value  of  n,  and  the  other  terms  the  effect  of  the 
increase  in  velocity  produced  by  molecular  attraction. 
The  first  term  corresponds  to  supposing  that  the  substance 
behaves  as  a  perfect  gas,  and  it  therefore  varies  inversely 
as  v  and  may  thus  be  written  C\/v,  where  Ci  is  a  constant 
which  may  be  determined  from  the  substance  in  the  gaseous 
state.  Of  the  remaining  terms  the  first  depends  on  the 
chance  of  each  molecule  coming  under  the  influence  of 
another  molecule,  which  varies  inversely  as  v2,  and  the 
second  term  depends  upon  the  chance  of  each  molecule 
coming  under  the  influence  of  two  molecules,  which  varies 


96  THE  EFFECT  OF  THE  MOLECULAR  FORCES 

inversely  as  v3,  and  so  on.     Thus  the  sum  of  these  terms 
may  be  written 


and  hence 

+  .  .  .Cz/v*.    .     .     (66) 


The  quantities  €2,  Cs,  .  .  .  Cz  are  evidently  functions  of 
v,  for  the  chance  of  two  molecules  coming  close  together 
will  be  influenced  by  the  forces  of  attraction  and  repulsion 
of  the  surrounding  molecules,  and  hence  €2  will  depend  on 
the  molecular  concentration,  or  on  v,  similarly  in  general 
the  chance  of  any  number  of  molecules  coming  close  together 
will  be  influenced  by  the  molecular  forces  of  the  surrounding 
molecules,  and  hence  €2,  C%,  .  .  .  Cz  will  be  functions  of 
v.  But  it  will  be  evident  on  reflection  that  the  variation  of 
n  with  v  in  the  foregoing  equation  will  in  the  main  be  ex- 
pressed by  the  inverse  powers  of  v,  in  other  words,  the 
quantities  €2,  €3,  .  .  .  Cz  are  functions  of  v  which  are  not 
sensitive  to  variations  of  v.  They  are  obviously  functions 
of  T  since  the  chance  of  a  number  of  molecules  coming 
close  together  will  depend  on  their  velocity  of  translation. 

It  follows  from  probability  considerations  that  the  terms 
C2/v2,  Cz/v3,  .  .  .  Cz/vz,  decrease  rapidly  in  magnitude  in 
the  order  they  stand.  For  the  chance  of  each  molecule  com- 
ing close  to  another  molecule  and  influencing  its  velocity 
of  translation  is  equal  to  a  fraction  pi,  and  the  chance 
of  each  pair  of  molecules  coming  close  to  a  third  molecule 
and  influencing  its  velocity  of  translation  is  evidently 
therefore  equal  to  pip2,  where  p2  is  a  fraction  which  is  prob- 
ably smaller  than  p\,  etc.,  and  accordingly 


If  the  quantities  C2,  Cs,  .  .  .  C2  be  expanded  in  inverse 
powers  of  v,  an  expression  for  n  is  obtained  which  may  be 
written 


AN  EXPRESSION  FOR  THE  QUANTITY  n  97 

.where  €20,  Csa,  -  -  -  Cza,  are  functions  of  T  only.  It  will 
be  evident  on  reflection  that  since  the  first  term  of  each 
expansion  will  be  large  in  comparison  with  the  remaining 
terms,  the  terms  C^a/v2,  Cza/v3,  .  .  .  C2a/vz  decrease  in 
magnitude  in  the  order  they  stand,  and  probably  rapidly. 

In  using  equation  (66)  we  may  as  a  first  approximation, 
from  what  has  gone  before,  omit  all  the  terms  on  the  right- 
hand  side  except  the  first  two.  Equation  (65)  then  assumes 
the  form* 


(67) 


since  Ci  =  — A/ — = —  according    to    equation   (21),    where 
ma  \     o 

A  =  5.087 X 10 ~20\/Tm  according  to  equation  (19),  a  denotes 
the  coefficient  of  expansion  and  /3  the  coefficient  of  com- 
pression per  dyne  at  the  absolute  temperature  T,  v  denotes 
the  volume  of  a  gram  molecule  of  the  substance  and  b  the 
apparent  volume  occupied  by  the  molecules,  ma  denotes 
the  absolute  and  m  the  relative  molecular  weight  of  a  mole- 
cule, and  R  the  gas  constant  whose  value  is  given  in  Section 
5.  If  the  distribution  of  molecular  velocities  is  taken  into 
account  the  right-hand  side  of  the  foregoing  equation  accord- 
ing to  Maxwell's  law  has  to  be  multiplied  by  1.085  (Section 
20). 

The  quantity  €2  we  have  seen  is  a  function  which  varies 
little  with  v  and  therefore  over  a  small  region  of  densities 
it  may  be  taken  as  constant.  Therefore  on  applying  equa- 
tion (67)  at  constant  temperature  to  a  substance  at  two 
densities  not  differing  much  from  each  other,  two  equa- 
tions are  obtained  from  which  €2  and  6  may  be  determined. 
The  values  of  n  corresponding  to  these  two  densities  may 
then  be  obtained  by  equation  (66)  retaining  two  terms 
only  of  the  right-hand  side.  Similarly  the  equation  may  be 
*  Not  previously  published  and  used. 


98 


THE  EFFECT  OF  THE  MOLECULAR  FORCES 


applied  to  another  pair  of  densities  not  differing  much  from 
each  other  and  from  the  othei  pair  of  densities,  and  the 
corresponding  values  of  n  calculated.  The  values  of  €2 
and  b  obtained  in  the  two  cases  are  not  necessarily  the  same, 
though  they  will  obviously  differ  very  little  from  each  other. 
Thus  by  a  repeated  application  of  equations  (67)  and  (66) 
the  values  of  n  and  b  may  be  obtained  for  different  densities 
of  a  substance.  The  accuracy  of  the  results  will  evidently 
increase  with  the  number  of  density  values  into  which  a 
given  region  of  densities  is  divided.  On  the  whole  the 
method  should  give  very  good  results. 

On  having  obtained  values  of  n  of  a  substance,  the  cor- 
responding values  of  Vt,  the  total  average  velocity  of  a 
molecule,  may  be  obtained  by  means  of  the  equation 

3rc 

Vt~Nc' 
given  in  Section  18. 

As  an  example  of  the  foregoing  investigation  the  values 
of  n  and  Vt  have  been  calculated  for  C02  at  0°  C.  correspond- 
ing to  the  pressures  100  and  200  atmospheres,  which  reduce 
the  substance  to  the  liquid  state,  the  results  being  given  in 
Table  VIII. 

TABLE  VIII 


CO2   AT   0°  C. 

p  in 
atmos. 

in  atmos. 

r  in  c.c.  per 
grm.  mol. 

n. 

v  cms. 

1 

100 
200 

1 

2730 
4586 

45.03 

42.90 

3.68  X1023 
7.17X1026 
7.80  X1026 

3.96  X104 
1.56X105 
1.62X105 

The  values  of  Pn  which  were  used  to  obtain  the  values 
of  Ta/0  by  equation  (48),  were  deduced  by  the  method  of 
Section  21  from  Amagat's  experimental  results  on  the 


THE  INCREASE  OF  Vt  WITH  THE  DENSITY         99 

relation  between  v  and  p.  Th0  Table  contains  the  values 
of  n  and  Vt  corresponding  to, a  pressure  of  one  atmosphere, 
in  which  case  the  substance  is  in  the  gaseous  state, 
the  values  of  the  foregoing  quantities  in  that  case  being 
given  by  equations  (21)  and  (35).  It  will  be  seen  that 
the  value  of  Vt  is  increased  to  about  four  times  its  original 
value  as  the  pressure  of  the  substance  is  increased  from  1 
to  100  or  200  atmospheres.  This  falls  into  line  with  the 
results  of  Sections/17,  21,  and  25,  according  to  which  the 
effect  of  molecular  attraction  on  the  molecular  velocity 
in  a  substance  in  the  liquid  state  should  be  quite  large. 
If  the  moleculajf-  forces  had  no  effect  on  the  magnitude  of 
the  motion  of  i  translation  of  the  molecules,  as  is  usually 
supposed,  the  v^alue  obtained  for  €2  should  have  been  zero, 
or  at  least  veryi  small.  The  result  conclusively  shows  that 
the  factor  Vt/Va  in  equation  (53)  cannot  be  put  equal  to 
unity,  and  that  the  right  hand  side  of  Van  der  Waals'  equa- 
tion of  state  must  be  modified  accordingly.  The  values  of 
n  and  Vt  obtained  may  be  corrected  according  to  Max- 
well's law  by  dividing  them  by  1.085. 

The  value  of  b,  the  apparent  molecular  volume  of  a 
gram  molecule  of  molecules  at  the  pressure  100  and  200 
atmospheres  is  equal  to  12.2  in  cc.,  and  thus  is  about  J  of 
the  total  volume  of  a  gram  molecule  of  the  substance.  It 
is  smaller  than  the  volume  at  absolute  zero,  which  according 
to  Section  24  is  approximately  J  the  volume  vc  at  the 
critical  point,  or  equal  to  25.5  cc.  This  difference  is  due, 
as  was  explained  in  Sections  19  and  24,  to  the  molecules 
getting  nearer  to  each  other  at  higher  temperatures  than  the 
absolute  zero  through  possessing  kinetic  energy  of  motion 
of  translation. 

If  the  value  of  the  right  hand  side  of  the  equation  (67) 
is  calculated  for  C02  at  a  temperature  0°C.  and  pressure 
of  50  atmos.,  using  the  foregoing  values  of  the  constant  b 
and  C2,  the  value  of  2562  atmos.  is  obtained.  This  agrees 


100       THE  EFFECT  OF  THE  MOLECULAR  FORCES 

well  with  the  value  of  2542  atmos.  obtained  directly  for  the 
left-hand  side  of  the  equation. 

This  investigation  may  be  extended  to  a  mixture  of 
substances.  In  the  case  of  a  mixture  of — say  molecules 
e  and  r,  we  would  have 

r   _ ..  ..  ..  i 

(68) 

according  to  equations  (41),  (45),  and  (48),  where  ne  denotes 
the  number  of  molecules  e  crossing  a  square  cm.  from  one 
side  to  the  other  in  the  mixture,  and  nr  has  a  similar  meaning 
with  respect  to  the  molecules  r.  Now  we  may  write 


2r  /£»n\ 

> (69) 


and 


similarly  as  before,  where  C\r  and  C\e  may  be  obtained  from 
the  substances  r  and  e  isolated  and  in  the  gaseous  state. 
The  resultant  equation  will  then  contain  the  four  variables 
Cf2r,  Cf2e,  V e  and  b'r,  which  may  be  determined  by  applying 
the  equation  to  the  mixture  at  constant  temperature  cor- 
responding to  four  densities  not  differing  much  from  each 
other.  Similarly  a  mixture  of  any  number  of  constituents 
may  be  treated. 

Another  method  may  be  used  depending  on  variations 
of  the  relative  concentration  of  the  constituents  of  the 
mixture  instead  of  variations  of  its  density  at  constant 
relative  concentration.  Thus  for  ne  and  nr  we  may  in  this 
case  write 

nr=ar'Nr+ar"N2r+a/"NrNe  } 
and  ,      .  "...    (71) 


CALCULATION  OF  MOL^CULAVrME!  .\     101 


where  a/,  a/',  a/",  a/,  a/',  and  a/",  are  constants  of  which 
a/  and  a/  may  be  determined  according  to  Section  11 
from  the  constituents  isolated  and  in  the  gaseous  state, 
and  Ne  and  Nr  denote  the  concentrations  of  the  molecules 
e  and  r  respectively.  Similarly  for  be  and  &/  we  may  write 


and  ,....     (72) 


where  ke,  kr,  and  ker  are  constants.  It  will  be  recognized 
that  ke  denotes  the  apparent  volume  due  to  the  interaction 
of  a  molecule  e  with  the  remaining  molecules  e,  and  kr  has 
a  similar  meaning,  while  Nrvekcr  denotes  the  apparent 
volume  due  to  the  interaction  of  a  molecule  e  with  the  mole- 
cules r,  or  vice  versa,  etc.  On  substituting  from  these  equa- 
tions in  equation  (68)  an  equation  is  obtained  containing 
seven  variables,  which  may  be  determined  by  applying  at 
constant  temperature  the  equation  to  seven  different  rela- 
tive concentrations  of  the  mixture  differing  little  from  each 
other. 

In  the  case  of  a  dilute  solution  we  may  evidently  write 

nr=arNr  } 
and  , (73) 


where  ar  anu  ae  are  constants.  The  foregoing  process  is 
thus  much  simplified  in  this  case. 

The  foregoing  method  may  be  simplified  by  using  the 
values  of  ber  and  6/  calculated  from  the  values  of  be  and  br 
of  the  constituents  in  the  pure  state;  or  we  might  make 
the  assumption  that  br'  =  be',  which  would  probably  intro- 
duce no  serious  error. 

In  Section  20  a  method  was  given  for  finding  approxi- 
mate values  of  be  and  b/  from  the  values  of  be  and  br  of  the 
constituents  in  the  pure  state.  Approximate  values  of 


102    :\f*I]S  £$Ffitrr«PF:  T^E  MOLECULAR  FORCES 


Cf2r  and  Cf2e  may  similarly  be  obtained,  and  hence  approxi- 
mate values  of  nr  and  ne  be  obtained  from  equations  (69) 
and  (70).  Let  CZT  stand  for  €2  in  equation  (67)  when  the 
substance  consists  of  molecules  r,  and  similarly  let  Cze  stand 
for  €2  when  the  substance  consists  of  molecules  e.  The 
quantity  C2r/vr2  then  depends  on  the  chance  of  each  mole- 
cule in  a  pure  substance  of  volume  VT  per  gram  molecule 
coming  close  to  another  molecule  and  changing  its  velocity 
of  translation.  This  dependence  may  be  exhibited  in  a 
certain  way  which  is  useful  in  dealing  with  mixtures.  The 
chance  of  each  molecule  to  get  close  to  another  molecule 
is  proportional  to  the  product  of  the  concentrations  of  the 
molecules,  or  proportional  to  NrNr.  Since  NTmaTVr=mr, 
where  mar  denotes  the  absolute  and  mr  the  relative  molec- 

ular weight  of  a  molecule,  we  have  Nr=—,  where  K  denotes 

VT 

a  constant.  Thus  the  foregoing  chance  is  proportional  tc 
l/vr2,  and  the  resultant  change  in  nr  expressed  by.  C2r/vr2. 
But  if  the  gram  molecule  of  molecules  r  contains  mole- 
cules e  besides,  a  molecule  r  may  have  its  velocity  changed 
through  coming  close  to  a  molecule  e.  The  chance  of  that 
happening  to  each  molecule  r  is  proportional  to  the  product 
of  the  concentrations  of  the  molecules  r  and  e,  or  proportional 
to  NgNr*  This  chance  according  to  the  foregoing  results  is 
proportional  to  l/vrve,  where  ve  denotes  the  volume  of  the 
mixture  containing  a  gram  molecule  of  molecules  e.  The 
resultant  change  in  the  value  of  nr  through  the  addition  of 


molecules  e  is  therefore  approximately  given  by 


\/C2rC 


where  Cze  and  Cir  refer  to  the  molecules  e  and  r  in  the  pure 
state.  The  term  Cf2r/vr2  in  equation  (69)  consists  of  the 
sum  of  the  foregoing  two  changes  in  nr  brought  about  by 
molecular  interaction,  or 


Cf2r  _  C2r   ,  V  Cir 
o  —      o  "T" 


o  —      o 
Vr2         Vr2  VTVe 


THE  INTERNAL  ENERGY  OF  A  SUBSTANCE        103 
Similarly  it  can  be  shown  that 


nT 

Ve2         Ve  VrVe 

The  values  of  nr  and  ne  obtained  from  equations  (69)  and 
(70)  by  the  help  of  equations  (74)  and  (75)  are  likely  to  be 
fairly  accurate.  The  accuracy  may  be  increased  by  using 
the  third  and  higher  power  terms  in  the  series  for  nr  and 
ne  in  the  equations. 

The  deduction  of  equation  (65)  used  in  this  Section 
involves  the  use  of  the  quantity  intrinsic  pressure.  This 
quantity  is  therefore  one  of  importance,  and  it  will  there- 
fore be  of  interest  to  obtain  further  relations  between  it 
and  other  quantities. 

30.  An  Equation  Connecting  the  Intrinsic  Pres- 
sure, Specific  Heat,  and  other  Quantities. 

If  U  denote  the  internal  energy  of  a  gram  molecule  of 
a  substance,  we  have  according  to  the  Differential  Calculus 
that 

(*U\   =  (8U\    (8v\       (5U 

\8TJP     \~~ 

The  term  on  the  left-hand  side  of  the  equation  is  the  specific 
heat  per  gram  molecule  at  constant  pressure  and  was  written 
Sipi  in  Section  13.  For  U  we  may  write 


(76) 


fts  in  Section  13,  where  Ua  denotes  the  potential  energy  of 
molecular  attraction   per  gram  molecule,   um  the  internal 


molecular  energy,  and  —  -  —  the  kinetic  energy  of  the  mole- 

cules when  situated  at  points  at  which  the  forces  of  the 
surrounding  molecules  neutralize  one  another,  this  energy 


104         THE  EFFECT  OF  THE  MOLECULAR  FORCES 


being  equal  to  the  kinetic  energy  in  the  gaseous  state  accord- 

ing to  Section  16.     According  to  Section  21,   -—  ?  =  PB  the 

ov 

intrinsic  pressure,  where  v  denotes  the  volume  of  a  gram 

molecule,  and  since  -(-r^)  =a  the  coefficient  of  expansion, 
v\ol  /  P 

the  equation  may  be  written 

/p    ,(8um\  \      .{&Ua\    ./dum\      3R  . 

SiPl=    P»+-  »«+-         +  +-      (77) 


The   quantity    l-r—  )      is   verv    probably  negligible   in 
\  ov  IT 

comparison  with  Pn,  and  may  therefore  be  omitted  from  the 
equation.     The  quantity   (-r^)    expresses    the  change  in 


potential  energy  of  molecular  attraction  of  a  substance 
with  change  in  temperature  at  constant  volume.  It  is  very 
likely  not  zero  in  the  case  of  a  substance  consisting  of  com- 
plex molecules,  since  according  to  Section  13  the  relative 
distribution  of  the  atoms  in  a  molecule  is  changed  with 
change  in  temperature,  which  would  give  rise  to  a  change 
in  the  forces  of  attraction  of  the  molecules  upon  each  other. 

For  the  same  reason  the  quantity  (—  ~]    is  not  likely  to 

be  zero  in  the  case  of  a  complex  substance. 

But  if  the  molecules  consist  of  atoms  it  is  highly  probable 

that  these  quantities  and  the  quantity  (-r^)     are    zero. 

\  dv  )T 

The  equation  then  becomes* 


(78) 


and  may  thus  be  used  to  calculate  Pn  in  such  a  case. 
*  Not  previously  published. 


THE  POLYMERIZATION  OF  SUBSTANCES  105 

If  the  value  of  Pn  for  a  substance  in  the  liquid  state, 
which  is  mon-atomic  in  the  gaseous  state,  is  calculated 
by  the  foregoing  equation,  and  it  does  not  agree  with  that 
given  by  equation  (48),  it  shows  that  the  substance  is 
polymerized  in  the  liquid  state.  Thus,  for  example,  equa- 
tion (78)  gives  the  value  of  8.8  X104  atmos.  for  Pn  for  Hg 
at  0°  C.,  while  equation  (48)  gives  1.25 X104  atmos.  It 
appears  therefore  that  Hg  is  polymerized  when  in  the  liquid 
state.  There  is  a  good  deal  of  other  evidence  pointing  to 
the  same  conclusion. 

Equation  (78)  may  also  give  approximate  values  of  Pn 
in  the  case  of  substances  consisting  of  complex  molecules, 
since  in  most  cases  the  differential  quantities  in  equation 
(77)  are  relatively  so  small  that  they  may  be  neglected. 
The  difficulty  of  applying  the  equation  to  a  substance  arises 
principally  through  being  dependent  on  a  knowledge  of 

o  r> 

the  nature  of  the  molecules,  since  the  quantities  —  and 

v  depend  upon  it.    In  the  case  of  a  partly  polymerized  sub- 
stance a  considerable  error  may  therefore  be  introduced. 

In  the  previous  Sections  we  have  considered  various 
properties  of  substances  which  do  not  depend  directly  on 
the  nature  of  the  motion  of  a  molecule  and  on  its  rapidity. 
There  are,  however,  some  properties  that  do,  and  which 
will  be  considered  in  the  next  Chapter.  These  properties 
are  directly  connected  with  the  recurring  nature  of  the 
path  of  a  molecule,  which  exists  according  to  the  laws  of 
probability.  This  introduces  the  idea  of  a  recurring  path 
of  average  length  associated  with  a  molecule,  which  will 
be  discussed  in  the  next  Section,  the  results  being  introductory 
to  the  matter  contained  in  the  next  Chapter. 


106        THE  EFFECT  OF  THE  MOLECULAR  FORCES 

31.  The  Mean  Free  Path  of  a  Molecule  under 
Given  Conditions. 

A  molecule  in  a  substance  has  its  velocity  continually 
changed  in  direction  and  magnitude  through  the  influence 
of  the  surrounding  molecules.  At  certain  points  along  its 
path  it  may  therefore  satisfy  certain  conditions  in  respect 
to  the  surrounding  medium.  The  straight  lines  joining 
these  points  will  be  called  the  molecular  free  paths  correspond- 
X  ing  to  the  given  conditions  .i  These  paths  will  obviously 
not  have  the  same  length,  from  probability  considerations, 
but  evidently  the  mean  free  path,  the  mean  length  of  a  large 
number  of  paths,  should  have  a  definite  magnitude. 

It  is  of  importance  and  interest  in  this  connection  to 
calculate  the  probability  that  a  molecule  will  pass  over  a 
distance  x  without  satisfying  the  conditions  of  the  free  path. 
This  was  first  done  by  Clausius  in  connection  with  the 
mean  free  path  of  molecular  collision.  The  investigation 
will  be  given  a  form  here  which  applies  to  any  kind  of  a  free 
path. 

The  probability  Px  that  a  molecule  will  pass  over  the 
distance  x  without  satisfying  the  path  conditions  may  be 
written 


and  the  probability  that  it  will  pass  over  the  path  x+  dx 
is  therefore 


But  this  is  also  according  to  the  rules  of  probability  equal 
to  the  product  of  'the  probability  Px  and  the  probability 
Pai  that  the  molecule  will  pass  over  the  distance  dx  without 
satisfying  the  path  conditions,  that  is 


PROBABILITY  OF  PATH  OF  GIVEN  LENGTH         107 

Now  if  I  denotes  the  mean  free  path  of  the  molecule,  the 
probability  of  it  satisfying  the  path  conditions  in  passing 
over  the  distance  dx  is  dx/l,  and  the  probability  of  it  not 
satisfying  the  path  conditions  is  therefore 

1-^=1* ; 

I 


Hence  the  preceding  equation  may  be  written 

-f  )-'• 

which  on  integrating  gives 


-  -'•+•* 


where  C  denotes  an  arbitrary  constant.  Since  the  probability 
that  the  molecule  will  not  satisfy  the  path  conditions  in 
passing  over  the  distance  x  =  0  is  unity,  it  follows  that  C=  1, 
and  hence 

Px=e~T,      ......     (79) 

which  expresses  the  probability  Px  that  the  molecule  will 
pass  over  the  distance  x  without  satisfying  the  free  path 
condition,  in  terms  of  the  quantities  x  and  the  mean  free 
path  I. 

The  number  nx  of  100  molecules  passing  over  the  distance 
x  without  satisfying  the  mean  free  path  conditions  is  given 
as  an  illustration  for  a  number  of  different  values  of  x  in 
Table  IX,  where  x  is  expressed  in  terms  of  the  mean  free 
path  I.  It  will  be  seen  that  a  molecule  rarely  passes  over 
•<*  distance  greater  than  its  mean  free  path. 

The  probability  that  a  molecule  has  a  free  path  lying 
between  the  lengths  x  and  x+dx  is  evidently 

P       p  —  _g  ~~  ~[    .  £^/£  (80) 

according  to  equation  (79). 


108        THE  EFFECT  OF  THE  MOLECULAR  FORCES 


TABLE  IX 


nx. 

X. 

nx. 

X. 

nx. 

X. 

99 

O.Oll 

78 

0.251 

14 

21 

98 

0.02Z 

72 

0.3331 

5 

31 

90 

0.1Z 

61 

0.51 

2 

41 

82 

0.21 

37 

1.  01 

1 

4.51 

In  Section  21  the  number  of  molecules  n  crossing  a  plane 
one  square  cm.  in  area  in  a  substance  was  investigated. 
In  connection  with  the  investigation  of  this  Section  it  is 
of  interest  to  determine  the  probability  that  the  path  of 
a  molecule  crossing  the  plane  has  a  length  lying  between  x 
and  x-\-dx.  It  is  evident  that  on  moving  the  plane  from  one 
position  to  another  parallel  to  itself  its  chance  of  cutting  a 
free  path  of  length  x  will  be  proportional  to  the  value  of  x. 
Therefore,  since  a  molecule  in  its  migrations  has  different 
consecutive  paths  the  probability  of  the  plane  cutting  one 
of  length  x  is  equal  to  the  ratio  of  x  to  the  sum  of  all  of  the 
different  paths  of  the  molecule  divided  by  their  number. 
The  latter  quantity  is  equal  to  the  mean  free  path  I,  and  hence 
the  ratio  is  equal  to  x/L  This  probability  must  be  multi- 
plied by  the  probability  that  the  molecule  has  a  path  lying 
between  x  and  x+dx  which  is  given  by  equation  (80). 
Therefore  the  probability  Pc  that  the  free  path  of  a  molecule 
which  lies  between  the  lengths  x  and  x+dx  cuts  the  plane 


is 


(81) 


It  should  be  noted  that  the  foregoing  investigation  is  inde- 
pendent of  the  nature  of  the  conditions  defining  the  free  path 
of  a  molecule  and  that  therefore  these  conditions  may  be 
given  any  form  we  please. 

It  will  be  of  interest  to  compare  the  square  of  the  aver- 


THE  MEAN  OF  SQUARES  OF  MOLECULAR  PATHS      109 

age  path  I,  or  I2,  with  the  mean  of  the  squares  of  the  differ- 
ent paths.    The  latter  quantity  is  given  by 


j^v  U/J(/  LJ\,     , 

I 

by  the  help  of  equation  (80).    Thus  the  latter  quantity  is 
double  the  former. 

We  are  in  a  position  now  to  develop  formulae  for  the 
quantities  viscosity,  conduction  of  heat,  and  diffusion, 
which  depend  directly  on  the  nature  of  the  motion  of  a 
molecule,  in  terms  of  quantities  referring  to  the  nature  of 
this  motion,  and  other  quantities.  In  order  that  the  formulae 
may  be  given  forms  applying  to  different  states  of  matter 
I  have  introduced  new  definitions  in  connection  with  the 
path  of  a  molecule  under  different  conditions,  which  make 
no  reference  to  molecular  collision,  and  apply  to  all  states 
of  matter.  They  lead  to  formulae  involving  quantities 
usually  not  considered  in  treatises  on  the  Kinetic  Theory 
of  Gases.  These  give  interesting  information  as  to  how  the 
various  molecular  properties  give  rise  to,  or  modify,  the 
viscosity,  conduction  of  heat,  and  diffusion,  of  a  substance. 
I  have  previously  made  an  attempt  to  obtain  formulae  for 
the  foregoing  properties  of  matter  independent  of  the  idea 
of  molecular  collision*,  and  finally  developed  the  results 
given  in  this  book,  which,  however,  differ  considerably 
from  the  previous  results,  and  are  therefore  more  or  less  new. 

*  Phil.  Mag.,  July,  1912,  pp.  101-118. 


CHAPTER    III 

QUANTITIES  WHICH  DEPEND  DIRECTLY  ON  THE  NATURE 
OF   MOLECULAR  MOTION 

32.  The  Coefficient  of  Viscosity  of  a  Substance. 

A  solid  in  contact  with  a  liquid  or  gas  at  rest  experiences 
a  resistance  when  displaced  depending  on  the  nature  of  the 
gas  or  liquid,  the  resistance  remaining  constant  when  the 
motion  is  uniform.  This  arises  through  matter  being  carried 
along  by  the  surface  of  the  solid.  This  property  of  liquids 
and  gases  is  known  as  viscosity.  It  can  be  shown  that 
it  primarily  arises  from  the  fact  that  the  molecules  of  a 
substance  undergo  a  rapid  motion  of  translation.  Thus 
suppose  that  n  molecules  per  square  cm.  per  second  strike 
the  lower  surface  of  the  solid  A  shown  in  Fig.  10,  which  is 
moving  with  a  velocity  V\  parallel  to  its  surface.  If  the 
average  momentum  parallel  tg/ the  surface  of  the  solid 
of  the  molecules  which  per  second  are  about  to  collide  with 
the  solid,  is  nV^rria,  the  momentum  after  collision  is  that 
corresponding  to  the  velocity  of  motion  of  the  solid,  or 
equal  to  nVima,  if  the  colliding  molecules  assume  the  velocity 
of  the  solid,  and  thus  the  solid  imparts  a  momentum  equal 
to  nma(Vi  —  ¥2)  to  the  substance  per  second.  According 
to  the  equation  Ft  =  maV,  this  momentum  is  equal  to  the 
force  F  that  has  to  be  applied  to  the  solid  per  square  cm. 
of  the  surface  in  order  to  maintain  uniform  motion,  or 
F  =  nma(Vi  —  ¥2)-  The  layer  of  liquid  in  the  immediate 
vicinity  of  the  surface  is  accordingly  set  in  motion,  and  this 

110 


VISCOUS  RESISTANCE  AND  VELOCITY  GRADIENT     111 

motion  is  communicated  to  the  next  layer,  and  so  on.  The 
whole  liquid  is  thus  set  in  motion  along  planes  parallel  to 
the  surface  of  the  solid,  the  velocity  of  motion  decreasing 
with  the  distance  from  the  surface. 

It  follows  from  the  above  equation  that  the  force  neces- 
sary to  maintain  a  velocity  Vi  of  the  solid  depends  upon  the 
value  of  FI  —  Fg,  which  is  proportional  to  the  velocity 


^ 


FIG.  10. 

gradient  of  the  liquid   in  the  immediate  vicinity  of  the 
surface,  and  thus  the  foregoing  equation  may  be  written 


=  nmaK 


dV 
5x 


(82) 


where  the  distance  x  is  measured  at  right  angles  to  the 
surface  of  the  solid,  and  K  denotes  a  constant.  If  the  velocity 
gradient  is  unity  the  corresponding  force  F  is  called  the 
coefficient  of  viscosity,  and  will  be  denoted  by  77.  Hence 
if  a  plane  moving  with  a  velocity  V\  is  at  a  distance  d  from 
a  fixed  plane,  the  force  F  that  has  to  be  applied  per  square 
cm.  of  the  moving  plane  may  be  written 


which  accordingly  becomes  equal  to  77,  when  the  velocity 
gradient  is  unity.    The  magnitude  of  the  viscosity  may  be 


112  THE  NATURE  OF  MOLECULAR  MOTION 

smaller  than  indicated  by  the  foregoing  investigation 
through  the  molecules  on  striking  the  moving  solid  not 
acquiring  exactly  its  velocity.  This  effect  is  known  as 
slipping  of  the  fluid  over  the  surface  of  contact  of  the  solid. 
Experiment  has  shown  that  there  is  no  appreciable  slipping 
in  the  case  of  liquids,  and  this  also  holds  in  the  case  of  gases, 
except  at  very  low  pressures.  This  point  will  be  further 
discussed  in  Section  34.  It  will  not  be  difficult  to  see  that 
if  the  moving  solid  exerted  a  repulsion  upon  the  molecules 
of  the  surrounding  fluid  this  would  have  the  effect  of  tend- 
ing to  turn  back  the  molecules  approaching  it  without 
giving  to  them  a  motion  parallel  to  that  of  the  surface  of 
the  solid.  In  that  case  slipping  would  occur.  It  is  not 
improbable  that  the  double  layer  of  electricity  which  exists 
at  the  interface  of  a  fluid  and  solid  may  give  rise  to  some 
slipping  in  this  way. 

The  coefficient  of  viscosity  of  a  substance  is  usually 
obtained  in  practice  by  measuring  the  volume  v\  of  the  sub- 
stance which  passes  per  second  through  a  narrow  tube  of 
radius  r  and  length  LI  under  the  action  of  a  pressure  p. 
If  the  substance  escapes  from  the  tube  with  a  velocity  which 
is  negligible  in  comparison  with  the  velocity  that  would  be 
obtained  with  a  tube  of  large  radius,  the  value  of  T\  according 
to  Poiseuille  is  given  by 


If,  however,  the  escaping  liquid  possesses  an  amount  of 
kinetic  energy  which  cannot  be  neglected  the  term  /-=r-  has 


to  be  subtracted  from  the  right-hand  side  of  the  foregoing 
equation  where  V  denotes  the  velocity  of  the  escaping 
fluid  and  p  its  density. 

The  viscosity  of  a  substance  depends  not  only  on  the 
velocity  of  translation  of  a  molecule,  but  on  the  nature  of 


THE  TRANSFER  OF  MOMENTUM  BY  A  MOLECULE     113 

the  motion,  and  it  will  therefore  be  necessary  to  define  a 
quantity  connected  with  the  path  described  by  a  molecule. 

33.  The   Viscosity  Mean  Momentum   Transfer 
Distance  of  a  Molecule  in  a  Substance. 

Consider  a  viscous  medium  in  motion  parallel  to  the  plane 
AB  which  is  kept  at  rest,  as  shown  in  Fig.  11.     Let  abc 


A  B 

FIG.  11. 

denote  the  path  of  a  molecule  moving  towards  the  plane 
A  B,  and  which  therefore  passes  progressively  into  slower 
moving  portions  of  the  substance.  The  molecule  during 
its  course  loses  momentum,  continually  but  in  varying 
amounts  to  the  medium  parallel  to  the  direction  it  is  moving, 
through  the  interaction  of  the  molecules  due  to  their  molec- 
ular forces  of  attraction  and  repulsion  and  their  molecular 
volume.  Similarly  a  molecule  passing  in  the  opposite 
direction  continually  acquires  momentum  from  the  medium. 
The  medium  evidently  does  not  acquire  or  lose  momentum 
only  at  isolated  points.  We  may  suppose,  however,  that 


114  THE  NATURE  OF  MOLECULAR  MOTION 

the  effect  is  equivalent  to  a  row  of  points  being  associated 
with  the  molecular  path,  not  necessarily  lying  on  it,  at 
which  only  the  medium  acquires  or  loses  momentum.  Fur- 
ther let  us  suppose  that  the  momentum  lost  by  the  medium 
at  a  point  is  equal  to  V^ma}  where  ¥2  denotes  the  velocity 
of  the  medium  at  the  point,  and  ma  the  molecular  weight 
of  a  molecule,  this  momentum  being  absorbed  by  the  migra- 
ting molecule,  while  the  momentum  gained  by  the  medium 
at  the  same  point  is  equal  to  V\ma,  where  Vi  denotes  the 
velocity  of  the  medium  at  the  preceding  point,  this  momen- 
tum being  abstracted  from  the  migrating  molecule.  The 
distance  between  two  consecutive  points  will  be  called  the 
momentum  transfer  distance  corresponding  to  the  path  of 
the  migrating  molecule. 

The  molecule  thus  appears  to  abstract  the  momentum 
V\ma  from  a  point  in  the  medium  and  to  transfer  it  to  a 
consecutive  point,  while  at  the  latter  point  it  abstracts 
the  momentum  V2ma  from  the  medium  and  transfers  it  to 
a  point  consecutive  to  the  latter,  and  so  on.  The  positions 
of  these  points,  and  their  number  per  unit  length  of  path, 
are  to  a  certain  extent  arbitrary,  as  will  easily  be  recognized, 
and  it  is  therefore  necessary  to  introduce  conditions  which 
will  render  the  lines  joining  them  quite  definite.  One  condi- 
tion that  the  points  obviously  have  to  satisfy  is  that  the 
flow  of  momentum  in  the  substance  should  be  uniform 
everywhere.  But  this  condition  alone  is  not  sufficient. 
Let  us  therefore  impose  the  two  additional  conditions  that 
the  number  n  of  molecular  paths  crossing  a  square  cm. 
from  one  side  to  the  other  shall  be  equal  to  the  number  of 
transfer  distances  passing  through  the  square  cm.,  and 
that  all  directions  of  a  transfer  distance  in  space  are  equally 
probable.  We  will  see  in  the  next  Section  that  these  con- 
ditions completely  determine  the  distribution  of  the  points 
in  question.  According  to  Section  31  the  various  transfer 
distances  corresponding  to  these  points  would  not  be  of 


RESULTANT  FORCE  ON  A  MOVING  MOLECULE     115 


the  same  magnitude,  but  grouped  about  a  mean  transfer 
distance  ln  according  to  a  general  law  whose  form  was  ob- 
tained. 

It  should  be  noted  that  according  to  the  definition  of  a 
transfer  distance  the  length  of  path  of  a  molecule  between 
two  points  is  equal  to  the  sum  of  the  corresponding  momen- 
tum transfer  distances.  The  transfer  distances  will  there- 
fore be  associated  with  the  molecular  path  in  the  way 
shown  in  Fig.  11. 

A  migrating  molecule  is  evidently  continually  undei  the 
action  of  the  resultant  force  arising  from  the  forces  of  the 


•Molecular  path 


FIG.  12. 


surrounding  molecules.  This  resultant  force  is  continually 
changing  in  direction  and  magnitude.  It  will  therefore  pass 
through  a  series  of  maxima  and  minima  of  different  mag- 
nitudes, which  may  graphically  be  illustrated  by  the  curve 
in  Fig.  12.  The  largest  maxima  correspond  to  the  closest 
possible  approach  of  two  molecules,  while  the  other  maxima 
correspond  to  various  distances  of  approach,  whose  effect 
is  modified  by  the  situation  of  the  surrounding  molecules. 
Thus  some  of  the  maxima  and  minima  will  have  very  small 
though  finite  values.  This  holds  for  a  gas  as  well  as  for  a 
liquid.  The  theoretical  points  at  which  a  molecule  is  sup- 
posed to  absorb  momentum  from  the  medium  or  transfer 


116  THE  NATURE  OF  MOLECULAR  MOTION 

momentum  to  it  evidently  therefore  do  not  necessarily 
correspond  to  maxima  or  minima  values  of  the  resultant 
force  acting  on  the  molecule,  and  besides  on  account  of  the 
small  values  of  some  of  the  maxima  and  minima  several 
of  these  are  likely  to  be  situated  between  two  such  consecu- 
tive points.  These  considerations  show  that  if  molecular 
forces  exist  between  molecules  the  momentum  free  path 
cannot  be  defined  with  respect  to  molecular  collision,  but 
must  be  defined  along  such  lines  as  developed  in  this  Section. 

It  will  not  be  difficult  to  see  that  even  if  the  molecules 
consisted  of  hard  elastic  spheres  not  surrounded  by  fields 
of  force  the  average  distance  between  two  consecutive  col- 
lisions in  a  gas  need  not  be  equal  to  the  length  of  the  aver- 
age momentum  transfer  distance  as  just  defined.  In  fact 
it  can  be  shown  that  these  distances  are  not  equal  to  one 
another.  But  the  opposite  is  usually  tacitly  implied  when 
the  mean  free  path  of  a  molecule  in  a  gas  is  defined  according 
to  molecular  collision  and  then  used  to  obtain  a  formula 
for  the  viscosity  of  a  gas  in  the  usual  way.  Of  course,  some 
mathematicians  have  begun  to  realize  this  and  endeavored 
to  determine  the  appropriate  factor  that  has  to  be  asso- 
ciated with  the  mean  free  path  in  the  formula.  But  this 
cannot  lead  to  anything  definite  because  the  interaction 
between  molecules  is  largely,  if  not  altogether,  due  to  the 
existence  of  the  molecular  forces. 

A  general  formula  for  the  viscosity  of  a  substance  in  the 
gaseous,  liquid,  and  intermediary  states  of  matter,  in  terms 
of  the  momentum  transfer  distance  will  now  be  developed 
and  applied  to  the  facts. 

34.  Formulae  for  the  Viscosity  in  Terms  of 
Other  Quantities. 

Let  us  consider  as  before  a  substance  moving  in  planes 
parallel  to  a  plane  AB  which  is  at  rest,  as  is  shown  in  Fig. 


THE  TRANSFER  OF  MOMENTUM  ACROSS  A  PLANE  117 


13.  Suppose  that  n  molecules  per  square  cm.  cross  the 
plane  EF  from  one  side  to  the  other.  The  migration  of 
these  molecules  gives  rise  to  a  transference  of  momentum 
across  the  plane  from  top  to  bottom  which  per  square  cm. 
is  numerically  equal  to  the  coefficient  of  viscosity  if  the 
velocity  gradient  is  equal  to  unity.  The  expression  for  this 
momentum  we  will  now  proceed  to  find.  Consider  the 
momentum  transfer  distances  which  cut  the  plane  EF  and 
are  inclined  at  the  angle  6  with  a  perpendicular  to  the  plane. 
If  we  suppose  for  purposes  of  calculation  that  these  distances 
start  from  the  same  point  which  we  will  suppose  lies  in  the 
plane  EF,  it  follows  from  the  figure,  which  shows  the  sec- 
Vl  >^ *- 


FIG.  13. 


tion  of  a  hemisphere  of  radius  z  made  by  a  plane  at  right 
angles  to  the  plane  EF,  that  the  number  of  these  distances 
is  equal  to 


where  z  denotes  the  length  of  one  of  these  distances,  r  =  z  sin  6, 
and  this  number  is  therefore  equal  to 

nsme-dd. 

If  nz  denote  the  number  of  the  foregoing  distances  whose 
lengths  lie  between  z  and  z+dz,  we  have 


nz 


z    —— 
n  sin  6  -  dO  •  --^e    *,  •  dzt 


118  THE  NATURE  OF  MOLECULAR  MOTION 

by  the  help  of  equation  (81),  where  Z,  denotes  the  mean 
momentum  transfer  distance.  The  molecules  corresponding 
to  these  distances  abstract  the  momentum  n2maVi  from 
the  plane  EF,  which  moves  with  the  velocity  Vi,  and  trans- 
fers it  to  the  plane  CD  which  moves  with  the  velocity  V2. 
From  considerations  of  equilibrium  it  follows  that  an  equal 
number  of  molecules  migrate  in  the  opposite  direction. 
These  abstract  the  momentum  nzmaV2  from  the  plane  CD 
and  transfer  it  to  the  plane  EF.  Thus  a  transference  of 
momentum  across  the  plane  takes  place  which  is  equal  to 

n,ma(Vl-V2). 

On  taking  the  velocity  gradient  equal  to  unity  we  have 
Vi-V2  = 

2COS0 

and  the  foregoing  expression  accordingly  becomes 
nzmaz  cos  0. 

To  obtain  the  total  momentum  transferred  per  square  cm. 
across  the  plane  EF,  or  the  coefficient  of  viscosity  77,  the 
foregoing  expression  on  substituting  for  nz,  must  be  integrated 
from  0  to  °o  with  respect  to  the  transfer  distance  z,  and 
from  0  to  7T/2  with  respect  to  the  angle  0,  giving 

rf  r  22  _L 

i)  =  nma  I       I     cos  0  •  sin  0  •  j-^e   lr,  -  d6  -  dz. 
Jo    Jo  ln 

Since 

ri 

cos  0' 


and 


JT< 


rv  -?- 

__/j        I-    .  fly  O7 

j  oe       ™     U"*  —  **»J 
JO       lr,2 


A  FORM  OF  THE  VISCOSITY  EQUATION  119 

the  foregoing  equation  becomes 

ri=nmalr,,     .......     (83) 

where  n  denotes  the  number  of  molecules  of  absolute  molec- 
ular weight  ma  crossing  a  plane  of  one  square  cm.  from  one 
side  to  the  other  per  second,  and  ln  denotes  the  mean  momen- 
tum transfer  distance. 

The  foregoing  equation  in  the  form  it  stands  is  mainly 
of  use  in  determining  the  value  of  Z,,  since  77  may  be  measured 
directly,  and  n  determined  by  the  method  described  in 
Section  29. 

The  quantity  ln  may  be  expressed  in  terms  of  other 
quantities.  Let  us  suppose  that  the  substance  is  represented 
by  another  substance  possessing  the  same  viscosity  and 
expansion  pressure,  but  whose  molecules  possess  no  volume 
of  the  kind  represented  by  the  symbol  6  (Section  19).  This 
is  possible  since  the  law  of  molecular  attraction  of  the 
representative  substance  is  initially  left  arbitrary,  and  may 
therefore  be  given  a  form  that  the  foregoing  conditions  are 
satisfied.  The  law  may  evidently  involve  as  many  arbitrary 
constants  as  we  please.  The  viscosity  of  the  representative 
substance  is  given  by 


and  l\  have  meanings  corresponding  to  n  and 
l^  in  equation  (83).  Since  the  representative  substance 
has  the  same  expansion  pressure  as  the  original  substance, 
and  the  molecules  of  the  former  substance  have  no  volume, 
we  have  from  equations  (38)  and  (45)  that 


v—b 
where  the  quantities  n,  v,  b,  Pn,  p,  and  A,  refer  to  the  original 


120  THE  NATURE  OF  MOLECULAR  MOTION 

substance,   and  the  preceding  equation  may  therefore  be 
written 

,.       •     •     •     (84) 


Each  quantity  in  this  equation,  except  l'n,  may  be  determined 
directly  or  indirectly  by  experiment,  and  the  latter  quantity 
is  therefore  determined  by  the  equation.  Tfye  fact  that 
the  equation  contains  a  variable  which  does  not  refer  directly 
to  the  original  substance  shows  that  it  may  be  represented 
by  another  substance  under  the  conditions  stated. 

On   comparing   the    foregoing    equation    with   equation 
(83)  we  see  that 


and  thus  /',,,  the  mean  transfer  distance  in  the  representative 
substance,  is  evidently  an  inferior  limit  of  lr  We  will  see 
later  that  /,  does  not  differ  appreciably  from  l\  except 
when  the  density  of  the  substance  corresponds  to  that  of 
the  liquid  state. 

It  is  often  of  interest  to  compare  the  distance  of  separa- 
tion d  of  the  molecules  in  a  substance  with  the  quantity 

Wi 
Z',.     We  have  immediately  d  =  —,  where  w\  is  a  constant. 

For  lrn  we  may  write  l\  =  W2V,  where  W2  is  a  function  of  v 
which  becomes  a  constant  when  the  substance  is  in  the 
gaseous  state.  The  ratio  of  l\  to  d  is  then  given  by 


(86) 


where  ws  is  a  function  of  v  which  becomes  a  constant  when 
the  substance  is  in  the  gaseous  state. 

The  quantity  I'    in  equation  (84)  may  be  expressed  in 


THE  INTERFERENCE  FUNCTION  OF  VISCOSITY      121 

terms  of  quantities  which  may  be  taken  to  refer  to  the 
original  substance.  If  the  molecules  in  the  representative 
substance  were  devoid  of  forces  of  attraction  and  repulsion 
which  extend  beyond  a  small  distance  from  each  molecule, 
Z',,  would  be  inversely  proportional  to  the  chance  of  one 
molecule  encountering  another,  and  thus  inversely  propor- 
tional to  the  concentration  of  the  molecules,  or  proportional 
to  the  volume  v,  and  thus  we  may  write 


under  these  conditions,  where  K^  is  a  constant  at  constant 
temperature.  The  existing  forces  of  attraction  and  repulsion 
have  the  effect  of  modifying  the  interaction  of  two  mole- 
cules, in  which  case  l'n  would  not  vary  according  to  the  fore- 
going equation.  The  effect  of  the  molecular  forces  may  be 
expressed  by  introducing  a  factor  into  the  right-hand  side 
of  the  foregoing  equation,  which  is  determined  from  the 
following  considerations.  The  chance  of  a  migrating  mole- 
cule coming  under  the  influence  of  another  molecule,  and 
the  pair  of  interacting  molecules  coming  under  the  influence 
of  a  third  molecule,  is  proportional  to  the  square  of  the 
molecular  concentration,  or  inversely  proportional  to  y2; 
the  chance  of  a  migrating  molecule  coming  under  the  influ- 
ence of  another  molecule,  and  the  pair  of  interacting  mole- 
cules coming  under  the  influence  of  two  molecules,  is  inversely 
proportional  to  v3;  and  so  on.  Thus  the  factor  in  question 
consists  of  the  series 


„    •     •     •     (87) 


where  0',/v2  expresses  the  average  effect  of  a  single  mole- 
cule on  a  pair  of  interacting  molecules,  (fr'^/v3  the  effect  of 
a  pair  of  molecules,  and  so  on,  while  3%  denotes  the  sum  of 
the  series  involving  v.  The  quantities  <£'n,  <£",,,  .  .  .  ,  are 


122          THE  NATURE  OF  MOLECULAR  MOTION 

functions  of  v  and  T,  which,  however,  probably  vary  little 
with  v.    For  I,,  and  l'n  we  have  therefore  the  expressions 


,     -     •     •     •     (88) 
and 


The  interpretation  of  equation  (88)  is  interesting. 
When  two  molecules  interact  the  process  is  modified  by  the 
surrounding  molecules,  which  effect  may  be  called  molec- 
ular interference.  The  value  of  b  depends  on  molec- 
ular interference,  but  probably  only  to  a  small  extent. 
But  the  quantity  <£,  depends  entirely  on  it,  and  this  quan- 
tity may  therefore  be  called  the  interference  function  in 
viscosity.  The  effect  of  molecular  interference  on  the 
value  of  Z,,  according  to  equation  (88),  is  expressed  by 
the  quantities  b  and  <£,,,  but  mainly  by  $,,  in  other  words, 
the  effect  of  molecular  interference  on  the  value  of  ln 
which  is  not  included  in  the  value  of  b  is  represented  by 
the  interference  function  <£,,,  which  may  now  be  used 
without  referring  to  the  representative  substance. 

Equation  (83)  referring  to  the  original  substance  may 
now  be  written  in  the  forms 

v2 

,      .....     (90) 


,    .     ..         .     (91) 

and 

•  .....     (92) 


by  means  of  equations  (35),  (38),  (45),  and  ^88),  remember- 
ing that  Ncmav  =  m,  the  molecular  weight  in  terms  of  that 
of  the  hydrogen  atom. 


THE  QUANTITIES  IN  THE  VISCOSITY  EQUATION     123 

The  values  of  n,  b,  and  Vt,  the  number  of  molecules 
crossing  a  square  cm.,  the  apparent  molecular  volume, 
and  the  total  average  velocity,  respectively,  may  be  deter- 
mined by  the  method  described  in  Section  29,  and  the  value 
of  the  intrinsic  pressure  Pn  by  the  method  described  in 
Section  21.  The  calculated  values  of  n.  b,  and  Vt  depend 
on  the  assumed  distribution  of  molecular  velocities,  *  and  may 
be  corrected  according  to  Maxwell's  law,  if  desired,  in  the 
way  described.  The  value  of  A  in  equation  (92)  not  cor- 
rected for  the  distribution  of  molecular  velocities  is  given  by 

A  =  5.087  X  W~20V^m 

according  to  Section  11.  If  corrected  according  to  Maxwell's 
law  the  value  has  to  be  multiplied  by  •*/-«-,  or  1.085.  It 

will  appear,  however,  from  Sub-section  a  that  the  calcu- 
lated value  of  K^  similarly  corrected  involves  the  factor  1.085, 
and  that  therefore  the  uncorrected  values  of  /c,  and  A  may 
be  used  in  equation  (92).  It  follows  then  that  in  equations 
(90)  and  (91)  the  uncorrected  values  of  Kn,  n,  b,  and  Vt 
may  be  used,  since  p  is  independent  of  molecular  motion 
(Section  8),  Pn  is  obviously  so  by  nature,  and  therefore 
the  factor  Kn/A  in  equation  (92)  is  represented  in  the  pre- 
ceding equations  by  factors  involving  *„  n,  6,  and  Vt. 

The  value  of  K,  for  a  given  substance  is  immediately 
determined  by  applying  equation  (92)  to  the  substance  in 
the  perfectly  gaseous  state,  which  corresponds  to  v  =  oo , 
and  therefore  to  Pn  =  0  and  <£>,,  =  0.  It  may  be  noted  that 
according  to  equations  (85),  (88),  and  (89)  we  have  1^  = 
/•/  =  «,«  when  the  original  and  representative  substances 
are  in  the  perfectly  gaseous  state,  and  the  value  of  Kn  is 
therefore  the  same  for  both  substances.  It  evidently  de- 

*This  distribution  refers  to  molecules  in  the  perfectly  gaseous 
state. 


124  THE  NATURE  OF  MOLECULAR  MOTION 

pends  on  the  molecular  forces  and  the  molecular  volume 
of  the  molecules  of  the  original  substance. 

It  will  be  evident  on  reflection  that  the  quantities 
n,  Vt,  6,  Pn,  and  p  are  affected  to  a  certain  extent  by  the 
influence  of  the  molecules  of  the  substance  on  two  molecules 
while  they  interact,  or  are  affected  by  molecular  interfer- 
ence. The  quantity  <£,,,  or  the  interference  function,  there- 
fore, represents  the  effect  of  molecular  interference  on  the 
viscosity  of  the  substance  which  is  not  represented  by  the 
foregoing  quantities  in  equations  (90),  (91),  and  (92). 
Thus  we  may  now  use  these  equations  without  referring  to 
the  representative  substance. 

The  series  for  the  interference  function  <£,  which  occurs 
in  the  foregoing  equations  may  be  given  a  simpler  form 
which  holds  approximately.  The  probability  of  a  migrat- 
ing molecule  coming  under  the  influence  of  another  mole- 
cule, and  the  pair  of  interacting  molecules  coming  under 
the  influence  of  y  molecules,  is  much  greater  than  the  prob- 
ability of  their  coming  under  the  influence  of  y+1  molecules, 

and  thus  the  terms  in  this  series  stand  in  the  order  of  mag- 

,'       i // 

nitude  —j>^->  •  -  •  ,  and  are  likely  to  decrease  in  mag- 
nitude very  rapidly.  As  a  first  approximation  we  may 

,  / 
therefore  retain  only  the  term  -^-,  or  replace  the  series  by 

$' 
the  term    .  ,  where  x  would  differ  little  from  2. 

It  is  instructive  to  apply  equation  (91)  to  a  hypothetical 
substance  whose  molecules  consist  of  perfectly  elastic  spheres 
not  surrounded  by  fields  of  force.  The  value  of  Vt  then 
corresponds  to  that  when  the  substance  is  in  the  gaseous 
state  at  the  same  temperature  according  to  Section  19, 
and  <£,,  represents  the  interference  by  actual  contact  of  one 
or  more  molecules  with  two  molecules  about  to  collide  or 
undergoing  collision.  Since  in  all  cases  $,,  in  the  representa- 


THE  VISCOSITY  OF  A  GAS  AND  ITS  DENSITY        125 

live  substance  represents  an  effect  due  to  molecular  attrac- 
tion, the  quantity  will  have  the  same  sign  in  this  particular 
case  as  found  in  practice,  namely  positive  according  to  Sub- 
section (d)  .  The  effect  of  the  true  molecular  volume  accord- 
ing to  equation  (91)  would  thus  be  to  increase  the  viscosity 
by  means  of  the  positive  apparent  volume  b,  and  the 
molecular  interference  represented  by  <£>,,,  to  both  of  which 
it  would  give  rise. 

The  number  Nt  of  times  the  mean  transfer  distance 
associated  with  the  path  of  a  molecule  is  passed  over  per 
second  is  a  quantity  of  interest.  Since  the  length  of  molec- 
ular path  between  two  points  is  equal  to  the  sum  of  the 
momentum  transfer  distances,  we  have  immediately 


by  the  help  of  equations  (83)  and  (35),  where  Vt  denotes 
the  total  average  velocity  of  a  molecule,  and  Ncma  =  p  the 
density  of  the  substance.  The  value  of  Vt  may  be  deter- 
mined by  the  method  of  Section  29  in  the  case  of  a>  liquid, 
or  gas  not  obeying  Boyle's  law. 

Applications  of  the  foregoing  equations  will  now  be  given. 

(a)  On  applying  equation  (90)  to  the  perfectly  gaseous 
state,  which  corresponds  to  v  =  co  ,  it  becomes 


,     . 
(94 


by  the  help  of  equation  (21).  It  follows  from  this  equation 
that,  since  K,  is  constant  at  constant  temperature,  77  is 
independent  of  the  density  of  the  gas.  This  remarkable 
result  was  first  deduced  by  Maxwell,*  and  has  been  amply 
confirmed  by  experiment.  The  deviations  that  have  been 
obtained  are  due  in  part  to  the  conditions  of  the  experi- 
*  This  deduction  depends  on  the  method  of  molecular  collision. 


126  THE  NATURE  OF  MOLECULAR  MOTION 

ment  not  conforming  to  the  conditions  underlying  the 
deduction  of  the  above  equation.  Thus  for  example,  the 
result  is  found  to  break  down  completely  at  extremely  low 
pressures.  Now  it  will  be  evident  from  an  examination 
of  the  deduction  of  the  foregoing  equation  that  it  will  hold 
only  so  long  as  Z,  is  small  in  comparison  with  the  thickness 
of  the  moving  layer  of  gas.  The  equation  would  therefore 
begin  to  break  down  when  the  value  of  l^  becomes  com- 
parable with  the  linear  dimensions  of  the  apparatus  used 
for  measuring  the  viscosity. 

There  is,  however,  another  cause  operating  giving  rise 
to  a  deviation  of  equation  (94)  from  the  facts  at  low  pres- 
sures. The  effect  of  the  slipping  of  the  gas  along  the  mov- 
ing plane  on  the  value  of  the  viscosity  increases  as  the 
pressure  decreases.  Thus  let  Va  denote  the  velocity  of  the 
moving  plane,  Vb  the  component  velocity  of  a  molecule 
parallel  to  the  plane  before  striking  it,  and  Vc  the  component 
velocity  of  rebound  parallel  to  the  plane.  Then  if  the  mole- 
cule undergoes  slipping  along  the  plane  during  rebound 
Va>Vc>Vb.  The  viscosities  t\  and  v\\  when  there  is  no 
slipping  and  slipping  respectively  are  proportional  to  the 
velocity  gradients  in  the  gas,  and  hence  proportional  to 
Va  and  Vc  the  corresponding  velocities  of  the  molecules 
on  rebound,  which  gives  ij/rji  =  Va/Vc.  Now  the  difference 
between  Va  and  Vb  increases  with  increase  of  the  mean 
momentum  transfer  distance,  or  distance  of  the  layer  of 
gas  from  which  the  molecule  comes,  and  thus  increases  with 
decrease  of  pressure.  The  difference  between  Va  and  Vc 
therefore  also  increases  with  decrease  of  pressure,  which 
increases  the  value  of  the  ratio  rj/rji  according  to  the  fore- 
going equation.  This  corresponds  to  a  decrease  of  771,  and 
thus  the  effect  of  slipping  becomes  the  greater  the  lower  the 
pressure. 

The  dynamical  mechanism  underlying  the  result  that 
the  viscosity  of  a  gas  is  independent  of  its  density  may  be. 


GASEOUS  VISCOSITY  AND  THE  TEMPERATURE      127 

illustrated  by  the  following  considerations.  A  molecule 
transfers  a  certain  amount  of  momentum  to  the  gas  at 
the  end  of  each  transfer  distance  on  migrating  at  right 
angles  to  the  motion  of  the  gas  in  the  direction  of  the  decrease 
of  motion.  If  the  concentration  of  the  gas  is  halved  the 
length  of  each  transfer  distance  is  doubled,  while  the  momen- 
tum transferred  at  the  end  of  each  transfer  distance  is  also 
doubled  since  the  velocity  gradient  of  the  gas  remains 
the  same.  Since  a  change  in  molecular  concentration  of  a 
gas  does  not  alter  the  molecular  velocities,  the  momentum 
transferred  per  second  by  a  molecule  moving  between  two 
parallel  plates  of  material  one  of  which  is  at  rest  while  the 
other  moves  parallel  to  itself  is  in  the  latter  case  double 
that  in  the  former.  But  since  the  number  of  molecules 
per  cubic  cm.  available  for  momentum  transference  in  the 
former  case  is  half  the  number  in  the  latter,  the  total  momen- 
tum transferred  is  in  each  case  the  same,  or  the  viscosity 
of  the  gas  has  not  been  altered  by  altering  its  density. 

The  quantity  K^  is  a  function  of  the  temperature.  This 
is  shown  by  the  calculated  values  of  K,  contained  in  Table 
X  for  a  number  of  gases  at  different  temperatures.  These 
values  are  not  corrected  for  Maxwell's  law  of  distribution 
of  velocities  since  n  given  by  equation  (21)  and  substi- 
tuted in  equation  (90)  was  not  thus  corrected.  If  this 
correction  is  carried  out  the  values  in  the  Table  have  to  be 


multiplied  by  -J-,  or  1.085.      It  will  be  seen  that  the  values 

of  /c,  increase  with  increase  of  temperature  in  the  case  of 
each  gas  mentioned  in  the  Table,  and  this  was  found  to  hold 
for  every  other  gas  that  was  examined.  This  indicates, 
since  the  mean  momentum  transfer  distance  /,,  is  equal  to 
KqV,  that  the  distance  over  which  a  migrating  molecule  trans- 
fers momentum  in  a  gaseous  medium  is  increased  by  an 
increase  of  temperature.  The  reason  probably  is  that  the 
greater  the  velocity  of  two  molecules  approaching  and 


128 


THE  NATURE  OF  MOLECULAR  MOTION 


receding  from  each  other  the  shorter  is  the  time  they  are 
under  the  influence  of  each  other's  attraction,  and  there- 
fore the  smaller  is  the  change  in  momentum  imparted  to 
each  other.  Large  changes  in  momentum  will  therefore 
take  place  only  for  close  distances  of  approach,  and  these 
will  therefore  take  place  less  frequently  along  the  path  of 
a  molecule  with  increase  of  temperature. 

TABLE  X 


MERCURY.     m  =  200.4 

KRYPTON,     m  =  81.8 

t°C. 

,10'. 

K,   10  ». 

t°C. 

r)  10'. 

if,  lO'o 

300 

380 

5320 
6560 

2.98 
3.94 

0 
100 

2334 
3063 

2.97 
3.33 

ARGON.     m  =  39.9 

HYDROGEN.     m  =  2 

-183.2 
0 
183.3 

735.6 
2104 
3243 

2.62 
3.83 
4.57 

-194.9 
0 
302 

374.2 

822 
1392 

5.69 
6.69 
7.81 

ETHER,    m  =  74 

ETHYL  CHLORIDE,     m  =  64.5 

0 
100 
212.5 

689 
967 
1234 

.92 
1.11 
1.24 

0 
157.3 
240.6 

935 
1440 
1714 

1.34 
1.64 
1.79 

CARBON  DIOXIDE.    m=44 

ETH  YLENE  .     m  =  28 

-21.5 
100 
302 

1278 
1972 
2682 

2.31 
2.92 
3.21 

-21.5 
99.25 
302 

891 
1278 
1826 

2.02 
2.38 
2.73 

M.      ISOBUTYRATE.      HI  =  102 

METHANE,     m  =  16 

24 
100 

754 
1122 

.823 
1.19 

0 
20 

1040 
1201 

2.99 
3.33 

THE  CHARACTERISTIC  CONSTANT  IN  VISCOSITY     129 

The  quantity  Kn  may  be  interpreted  in  other  ways. 
17  represents  the  rate  at  which  a  plane  at  right  angles  to  a 
unit  velocity  gradient  loses  momentum  per  square  cm. 
per  second.  Since  n  molecules  cross  the  plane  per  square 
cm.  per  second  in  each  direction,  rj/n  represents  the  momen- 
tum conveyed  by  a  molecule  across  the  plane  on  crossing 
it  and  recrossing  it  (sometime)  in  the  opposite  direction. 
According  to  equation  (90)  this  momentum  is  equal  to 
K^niaV  in  the  case  of  a  gas.  Thus  K,  represents  the  momen- 
tum conveyed  per  unit  mass  of  the  molecule  at  unit  volume 
of  a  gram  molecule  of  the  gas.  Hence  for  substances  of 
equal  K^  and  the  same  molecular  concentration  the  momen- 
tum conveyed  by  a  molecule  is  proportional  to  its  mass. 

A  molecule  crossing  the  plane  gets  a  distance  away 
from  it  which  on  the  average  is  proportional  to  the  volume 
v,  *  and  the  molecule  therefore  tal:e&  a  time  proportional  to 
v/Va  in  crossing  and  recrossing  the  plane,  where  Va  denotes 
its  average  velocity.  The  amount  of  momentum  conveyed 
across  the  plane  per  second  by  the  same  molecule  is  there- 
fore proportional  to  K,,maya,  and  thus  independent  of  the 
volume  of  the  gas. 

The  quantity  K^  according  to  equation  (91)  applied 
to  the  gaseous  state,  is  a  measure  of  the  total  momentum 
parallel  to  the  motion  of  the  gas  conveyed  per  square 
cm.  per  second  across  the  plane  per  unit  momentum  of  the 
momentum  of  translation  motion  of  a  molecule. 

A  number  of  formulae  of  an  empirical  nature  expressing 
the  variation  of  I,,,  or  K^V,  with  T  at  constant  v,  have  been 
given,  but  which  need  not  concern  us  here.f  It  is  useful 

*  This  distance  is  evidently  proportional  to  the  chance  of  the  path 
of  the  molecule  not  undergoing  a  deflection  per  unit  length,  which  is 
proportional  to  I,,,  and  hence  proportional  to  v,  since  I^^K^V. 

t  In  these  investigations  the  quantity  ln  is  supposed  to  represent 
molecular  free  path  according  to  the  method  of  collisions,  which  is 
connected  with  the  viscosity  by  the  equation  given  at  the  end  of  this 
Sub-section . 


130 


THE  NATURE  OF  MOLECULAR  MOTION 


to  note,  however,  that  ln  and  Kn  are   roughly  proportional 
to  the  square  root  of  the  absolute  temperature. 

The  viscosity  coefficient  of  a  gas  appears  to  obey  approxi- 
mately the  relation  of  corresponding  states,  and  this  holds 
therefore  also  for  the  quantity  K^  according  to  equation 
(94).  This  is  shown  by  Table  XI,  which  gives  the  ratio 

TABLE  XI 


Substance. 

Tc. 

27V 

•nc  107. 

772C   107. 

roe 
T,c 

Kr,c  1010. 

C02 

304.9 

609.8 

1625 

2811 

1.73 

2.66 

C2Ht 

282 

564 

1022 

1802 

1.76 

2.19 

A 

152 

304 

1216 

2298 

1.89 

2.97 

N2O 

318.4 

637.8 

1539 

2885 

1.88 

2.41 

H2 

32 

64 

272 

645 

2.36 

6.46 

of  the  viscosities  corresponding  to  the  temperatures  Tc 
and  2TC  for  a  few  substances,  where  Tc  denotes  the  critical 
temperature,  which  were  interpolated  from  the  viscosity 
data  given  in  Landolt  and  Bornstein's  Tables,  4th  edition. 
The  ratios,  it  will  be  seen,  are  approximately  equal  to  each 
other.  The  somewhat  large  deviation  in  the  case  of  H2  is 
probably  due  to  the  greater  uncertainty  attached  to  the  data 
in  that  case,  and  to  a  lack  of  sufficiently  extensive  data 
for  reliable  interpolation.  The  deviation  is,  however,  small 
in  comparison  with  the  difference  between  the  viscosity 
of  H2  and  that  of  each  of  the  other  substances.  The  varia- 
tion of  Kn  with  the  temperature  is  therefore  approximately 

expressed  by  a  function  of  the  form  «  c.^(-— )    where  K_C 

\TcJ 

denotes  the  value  of  K,,  at  the  critical  temperature,  and  which 
is  thus  a  fundamental  and  characteristic  quantity  of  a  gas. 
Its  values  for  a  few  substances  are  given  in  Table  XI,  ob- 
tained in  the  same  way  as  the  values  given  in  Table  X. 


THE  FREE  PATH  AND  MOLECULAR  SEPARATION     131 

The  value  of  /,,  for  a  substance  in  the  gaseous  state  at 
standard  temperature  and  pressure  is  obtained  by  multi- 
plying the  corresponding  value  of  *,,  by  the  volume  of  a 
gram  molecule  of  the  substance  under  standard  conditions, 
which  according  to  Avogadro's  law  has  the  same  value  for 
all  substances  and  is  equal  to  22,700  cc. 

It  is  often  of  interest  to  compare  the  value  of  ^  with  the 
average  distance  of  separation  d  of  the  molecules,  which  is 

Im  v\l/i     I        v        \A 
given  by  d  =  (  —  -  )    =  U-          231    ,  and  the  ratio  of  these 

quantities  is  therefore  given  by 


(95) 


It  will  be  found  that  on  substituting  for  Kn  and  v  at  standard 
temperature  and  pressure  that  ln  is  considerably  larger  than 
d,  showing  that  except  for  distances  of  approach  of  two 
molecules  much  less  than  d  the  effect  of  their  interaction  is 
small. 

If  equation  (91)  is  applied  to  the  gaseous  state,  and  ln 
is  written  for  Knv  according  to  equation  (88)  applied  to  the 
gaseous  state,  the  equation  assumes  the  form 


IV  .ml,     1 


where  —  =  p  the  density  of  the  gas.    This  equation  is  usually 

obtained  in  treatises  on  the  Kinetic  Theory  of  Gases  on 
the  supposition  that  each  molecule  has  the  same  velocity 
Vt,  where  Z,  is  supposed  to  denote  the  mean  free  path  between 
consecutive  collisions  of  a  molecule.  Since  /„  cannot  repre- 
sent exactly  this  quantity  the  equation  has  sometimes 
been  modified  by  the  introduction  of  an  appropriate  numer- 
ical factor  as  pointed  out  in  Section  33.  Another  factor  is 
introduced  on  taking  into  account  the  distribution  of  molec- 


132 


THE  NATURE  OF  MOLECULAR  MOTION 


ular  velocities  since  this  affects  the  chance  of  collision. 
This  represents  the  most  that  has  been  achieved  in  the  way 
of  a  Kinetic  Theory  of  substances  in  connection  with  vis- 
.cosity  according  to  the  idea  of  molecular  collision. 

(6)  It  will  be  of  interest  to  consider  the  values  of  l\ 
of  some  substances  in  the  liquid  state,  since  they  are  inferior 
limits  of  \,  from  which  they  differ  but  little,  and  they  can 
usually  be  more  easily  obtained  than  the  values  of  ln.  Table 
XII  gives  the  values  of  Z',  calculated  by  means  of  equa- 

TABLE  XII 


ETHER 

t°c. 

r  108  cm. 

d  108  cm. 

£ 

d  ' 

13.5 

.001779 

2.06 

5.49 

.376 

25.4 

.001649 

2.08 

5.52 

.377 

63 

.001338 

2.21 

5.65 

.392 

78.5 

.001241 

2.35 

5.70 

.412 

99 

.001133 

2.91 

5.80 

.502 

BENZENE 

15.4 

.004387 

3.25 

5.22 

.623 

50.1   • 

.003641 

3.07 

5.29 

.581 

78.8 

.003000 

3.60 

5.36 

.486 

CHLOROFORM 

0 

.003827 

3.05 

5.01 

.607 

20 

.003419 

2.84 

5.06 

.562 

40 

.003073 

2.73 

5.10 

.535 

60 

.002791 

2.68 

5.15 

.520 

tion  (84)  for  a  number  of  substances  in  the  liquid  state  at 
different  temperatures,  and  for  comparison  the  corresponding 
values  of  d  the  average  distance  of  separation  of  the  mole- 


INFERIOR  LIMITS  OF  THE  FREE  PATH 


133 


cules.  The  value  of  A  used  in  the  equation  was  not  cor- 
rected for  the  distribution  of  molecular  velocities.  If  cor- 
rected according  to  Maxwell's  law  the  values  of  l\  in  the 


Table  have  to  be  multiplied  by          ,  or  1.085.     Both  the 

quantities  I  \  and  d  have  values  of  the  order  of  magnitude 
10  ~8  cm.  The  latter  values  are  greater  than  the  former, 
as  is  indicated  by  the  values  of  the  ratio  I'^/d.  The  values 
of  Pn  (which  must  be  reduced  to  dynes)  used  in  the  cal- 
culations are  contained  in  Table  III,  while  the  values  of 
f\  were  calculated  by  means  of  the  empirical  formula  of 
Thorpe  and  Rodgers.* 

Table  XIII  gives  the  values  of  Z',  for  CO2  at  40°  C. 
under  high  pressures  in  the  gaseous  state,  and  the  correspond- 
ing values  of  d.  According  to  equation  (85),  taking  the 

TABLE  XIII 


CO2  AT  40°  C. 

p  in 
atmos. 

1'^  108  cm. 

d  108  cm. 

p  in  atmos. 

l'n  108  cm. 

d  108  cm. 

70 

6.52 

7.34 

94 

2.84 

5.16 

80 

4.55 

6.52 

100 

2.83 

5.03 

85 

3.82 

5.95 

112 

2.81 

4.90 

value  of  b  for  CO2  obtained  in  Section  29,  the  value  of  I',, 
is  smaller  than  the  value  of  I,,  for  the  greatest  density  by 
about  16  per  cent.  Thus  ^  and  I' \  do  not  differ  much  from 
each  other.  The  values  of  Pn+p  and  17  used  in  these  calcu- 
lations are  given  by  Table  XIV. 

(c)  The  number  of  times  Nt  that  the  mean  momentum 
transfer  distance  of  a  molecule  in  a  substance  is  passed  over 
per  second,  given  by  equation   (93),  is  a  quantity  of  inter- 
est.   This  number  for  C02  in  the  liquid  state  at  0°  C.  under 
*  Phil.  Trans.,  A.,  1894,  p.  1. 


134         THE  NATURE  OF  MOLECULAR  MOTION 


a  pressure  of  100  atmos.  is  6X1012,  where  the  value  of  Vt 
was  obtained  from  Table  VIII,  ??  =  . 000925,  and  p=.87. 
The  number  obtained  is  mainly  of  interest  on  account  of 
its  great  magnitude.  It  is  interesting  to  note  that  this 
number  is  smaller  than  the  number  of  times  per  second  the 
resultant  force  acting  on  a  molecule  due  to  the  surrounding 
molecules  passes  through  a  maximum  or  minimum. 

(d)  Let  us  next   apply   equation    (92)   to  some  of  the 
facts  to  obtain  values  of  <£,,.    Table  XIV  contains  values  of 

TABLE  XIV 


CO2  AT  40°  C.     *„  =  2.578  X  10-  10 

P  in 

Pn+P 

v  per 

r,  10« 

*n 

4.03X103 

atmos. 

in  atmos. 

grm.  mol. 

V  2-12 

70 

128.7 

245.7 

200 

80 

200.9 

172.2 

218 

.027      . 

.073 

85 

295.9 

130.7 

269 

.132 

.132 

94 

611.2 

85.35 

414 

.292 

.325 

100 

719.3 

78.96 

483 

.385 

.385 

112 

854.3 

73.24 

571 

.486 

.448 

<£,,  for  C02  at  different  pressures  at  a  temperature  of  40°  C., 
at  which  the  gas  does  not  assume  the  liquid  state  however 
great  the  pressure.  The  values  of  Pn+p  used  where  cal- 
culated by  means  of  Van  der  Waals'  equation  of  state  (Sec- 
tion 26)  writing  for  b  the  value  42.8  cc.  per  gram  molecule, 
which  results  from  the  conditions  expressed  by  equations 
(57)  and  (58).  The  value  corresponding  to  a  pressure  of 
70  atmos.  thus  obtained  is  very  nearly  equal  to  that  given 
in  Table  XVI,  which  was  obtained  by  a  different  method. 
The  values  obtained  for  Pn+p  are  therefore  likely  to  be  at 
least  approximately  correct.  The  values  of  77  used  are  those 
obtained  by  Phillips*.  Equation  (94)  gives  the  value  of 
*  Proc.  Roy.  Soc.,  A.,  Vol.  LXXXVII,  pp.  56-57. 


INFERIOR  LIMITS  OF  THE  FREE  PATH 


133 


cules.  The  value  of  A  used  in  the  equation  was  not  cor- 
rected for  the  distribution  of  molecular  velocities.  If  cor- 
rected according  to  Maxwell's  law  the  values  of  I',,  in  the 


Table  have  to  be  multiplied  by 


, 


or  1.085.     Both  the 


quantities  l\  and  d  have  values  of  the  order  of  magnitude 
10  ~8  cm.  The  latter  values  are  greater  than  the  former, 
as  is  indicated  by  the  values  of  the  ratio  I'Jd.  The  values 
of  Pn  (which  must  be  reduced  to  dynes)  used  in  the  cal- 
culations are  contained  in  Table  III,  while  the  values  of 
77  were  calculated  by  means  of  the  empirical  formula  of 
Thorpe  and  Rodgers.* 

Table  XIII  gives  the  values  of  £'„  for  C02  at  40°  C. 
under  high  pressures  in  the  gaseous  state,  and  the  correspond- 
ing values  of  d.  According  to  equation  (85),  taking  the 

TABLE  XIII 


CO2  AT  40°  C. 

p  in 
atmos. 

l'^  108  cm. 

d  108  cm. 

p  in  atmos. 

Vq  108  cm. 

d  108  cm. 

70 

6.52 

7.34 

94 

2.84 

5.16 

80 

4.55 

6.52 

100 

2.83 

5.03 

85 

3.82 

5.95 

112 

2.81 

4.90 

value  of  6  for  C02  obtained  in  Section  29,  the  value  of  l\ 
is  smaller  than  the  value  of  l^  for  the  greatest  density  by 
about  16  per  cent.  Thus  ^  and  V ^  do  not  differ  much  from 
each  other.  The  values  of  Pn+p  and  17  used  in  these  calcu- 
lations are  given  by  Table  XIV. 

(c)  The  number  of  times  Nt  that  the  mean  momentum 
transfer  distance  of  a  molecule  in  a  substance  is  passed  over 
per  second,  given  by  equation   (93),  is  a  quantity  of  inter- 
est.   This  number  for  CO2  in  the  liquid  state  at  0°  C.  under 
*  Phil.  Trans.,  A.,  1894,  p.  1. 


136          THE  NATURE  OF  MOLECULAR  MOTION 

stant.  The  foregoing  considerations  indicate  the  fundamental 
reasons  for  the  magnitude  of  the  value  of  the  viscosity  of  a 
substance,  and  the  reason  why  its  value  increases  rapidly 
with  the  density  when  the  substance  does  not  behave  as  a 
perfect  gas. 

Interesting  information  may  now  also  be  obtained  on 
the  effect  of  molecular  interference  on  the  motion  of  a 
molecule. 

The  positive  nature  of  <!>,,  indicates,  according  to  equa- 
tion (89),  that  the  effect  of  molecular  interference  is  to 
increase  l'n  from  the  value  it  would  have  if  it  varied  pro- 
portionally to  v,  which  corresponds  to  the  absence  of  molec- 
ular interference.  The  same  remark  applies  to  /,,  which  is 
given  by  equation  (88),  since  an  increase  of  b  would  be 
attended  by  an  increase  of  molecular  interference.  This 
signifies  that  the  chance  of  momentum  being  transferred 
by  a  migrating  molecule  to  another  molecule  when  moving 
along  the  velocity  gradient  of  the  substance  is  reduced  by 
the  vicinity  of  other  molecules  through  the  interaction  of 
their  forces  of  attraction  and  repulsion.  But  since  the  excess 
of  momentum  of  the  migrating  molecule  must  eventually 
be  transferred  to  the  medium  the  act  of  transference  when 
molecular  forces  exist  is  less  frequent,  but  when  it  occurs 
more  momentum  is  transferred,  than  would  be  the  case  in 
the  absence  of  molecular  forces.  In  the  former  case  when 
a  molecule  transfers  momentum  to  the  medium  the  differ- 
ence between  its  velocity  and  that  of  the  medium  in  the 
direction  the  medium  is  moving  is  greater  than  in  the  latter 
case,  and  hence  we  would  expect  that  the  amount  of  momen- 
tum transferred  would  be  greater. 

Assuming  that  the  series  <£,  may  approximately  be  rep- 
resented by  the  term  0,/w*  the  values  of  </>,  and  x  cor- 
responding to  the  data  in  Table  XIV  were  calculated  from 
the  values  of  $,,  corresponding  to  the  pressures  85  and 
100  atmos.,  giving  the  values  4.03  X 103  and  2.12  respectively. 


VALUES  OF  THE  INTERFERENCE  FUNCTION       137 


The  value  of  x  does  not  differ  much  from  2  as  was  pre- 
dicted. The  last  column  of  the  Table  contains  the  values 
of  <t>r,/vx  for  different  pressures,  calculated  by  means  of  the 
foregoing  values  of  0,  and  x.  They  agree  fairly  well  with 
the  values  of  $>,,  in  the  preceding  column.  A  better  agree- 
ment would  evidently  have  been  obtained  if  <£,  had  been 
taken  a  function  of  v  which  decreases  with  increase  of  v. 

Table  XV  gives  the  values  of  $,  at  different  temperatures 
for  a  few  liquids  (whose  density  depends  of  course  on  the 

TABLE  XV 


ETHER 

t°C. 

Gas 
r,  10'. 

KTI  lo1". 

*,. 

0,  io4. 

13.5 

723 

.9612 

1.09 

1.148 

99 

942 

1.074 

1.235 

1.857 

CHLOROFORM 

o 

959 

1.028 

2.783 

1.707 

60 

1 

1163 

1.129 

1.802 

2.011 

BENZENE 

15.4 

755 

.974 

2.772 

2.158 

78.8 

1079 

1.263 

1.154 

1.058 

temperature),  calculated  by  means  of  equation  (89),  using 
the  values  of  /'„  contained  in  Table  XII.  The  values  of 
KT,  were  calculated  by  means  of  equation  (94)  using  the 
values  of  77  corresponding  to  the  gaseous  state  given  in  the 
second  column  of  the  Table.  It  will  be  seen  that  the  values 
of  $,,  in  some  cases  increase  with  increase  of  temperature, 


138  THE  NATURE  OF  MOLECULAR  MOTION 

which,  it  should  be  noted,  is  attended  by  an  increase  of 
the  volume,  while  in  the  other  cases  the  values  decrease. 
Thus  the  effect  of  an  increase  of  the  temperature  on  the 
value  of  $„  appears  to  be  in  the  opposite  direction  to  the 
effect  of  an  increase  of  the  volume.  If  the  quantity  ^ 
behaves  similarly  for  all  substances,  as  we  might  expect, 
the  foregoing  results  would  indicate  that  at  constant  tem- 
perature it  would  decrease  with  increase  of  volume,  as  the 
preceding  results  have  already  shown,  and  increase  with 
increase  of  temperature  at  constant  volume.  This  indicates 
that  the  chance  of  transfer  of  momentum  by  a  migrating 
molecule  to  a  molecule  of  the  surrounding  medium  through 
the  existence  of  molecular  forces  is  decreased  by  an  increase 
of  temperature. 

Table  XV  also  contains  the  values  of  </>,,  in  the  approxi- 
mate expression  of  ^/vx  for  $„,  putting  x  =  2.  The  values 
of  77  for  different  volumes  at  constant  temperature  of  the 
substances  mentioned  in  the  Table  might  now  be  approxi- 
mately calculated  by  means  of  equation  (92)  on  calcula- 
ting the  values  of  Pn+p  by  means  of  an  empirical  equation 
of  state  according  to  Section  21. 

Further  investigation  in  connection  with  the  experi- 
mental values  of  77  of  liquids  must  proceed  mainly  along  the 
line  of  comparing  the  values  of  the  characteristic  quantity 
$  of  different  substances  at  various  temperatures  and  vol- 
umes. This  might  furnish  some  information  as  to  how  this 
quantity  depends  on  the  molecular  weight  besides  on  the 
volume  and  temperature,  and  incidentally  furnish  further 
information  in  connection  with  the  molecular  forces. 

(e)  It  is  interesting  to  apply  equation  (91)  to  a  substance 
not  in  the  perfectly  gaseous  state  at  volumes  for  which  6 
is  small  in  comparison  with  i>,  and  <!>,,  small  in  comparison 
with  unity,  in  which  case  the  equation  becomes 

(96) 


EFFECT  OF  DENSITY  ON  MOLECULAR  VELOCITY      139 

Corresponding  to  the  smallest  value  of  v  for  which  these 
conditions  hold  rj  usually  differs  considerably  from  that 
applying  to  the  substance  in  the  perfectly  gaseous  state, 
and  this  therefore  also  holds  for  Vt  and  n.  The  value  of 
Vt  may  therefore  be  calculated  with  fair  accuracy  by  means 
of  the  foregoing  equation  over  a  region  at  the  beginning  of 
which  a  substance  begins  to  deviate  from  the  gas  laws. 
The  corresponding  values  of  Pn+p,  and  hence  of  Pn,  may 
be  obtained  from  equation  (92).  Table  XVI  contains  a 


TABLE  XVI 


CO2  AT  40°  C.     Kr,  =  2.578  X  1.085  X  10~  10 

p  in 
atmos. 

77x106. 

»  per 
grm.  mol. 

Pn+P 

in  atmos. 

Pn™ 
atmos. 

Vt/Va- 

1 

157         25,550 

1 

1.000 

40 

176 

589.8 

48.59 

8.6 

1.121 

60 

187 

328.7 

92.70 

32.7 

1.191 

70 

200 

245.7 

132.5 

62.5 

1.274 

set  of  calculations  carried  out  for  CCb  at  40°  C.  over  a  range 
of  volumes  for  which  the  foregoing  conditions  hold,  since 

4  03  X 103 
approximately  6=12,  Section  29,  and  $,,=  - ' 


.12 


accord- 


ing to  Table  XIV.  The  value  of  K^  in  equation  (96)  was 
determined  by  applying  equation  (94)  to  C02  in  the  gaseous 
state  at  a  pressure  of  one  atmos.  and  temperature  40°  C., 
and  correcting  the  value  obtained  according  to  Maxwell's 
law.  It  will  be  seen  that  Pn  and  the  ratio  Vt/Va,  where  Va 
denotes  the  average  velocity  of  translation  of  a  molecule 
of  CO2  in  the  gaseous  state,  which  is  given  by  equation 
(8)  and  the  equation  Fa=.922F,  gradually  increase  with 
increase  of  pressure.  Thus  the  more  the  molecules  come 
under  each  other's  influence  the  greater  the  total  average 


140  THE  NATURE  OF  MOLECULAR  MOTION 

velocity  above  that  in  the  gaseous  state,  which  falls  into 
line  with  what  has  been  obtained  before.  It  is  of  impor- 
tance to  notice,  however,  that  the  foregoing  deduction  of  the 
result  does  not  depend  on  the  results  of  Sections  16  and  17. 

(/)  On  using  the  term  ^/v2  for  the  series  $,,  in  equation 
(90),  which  holds  approximately  according  to  Sub-section 
d,  and  substituting  for  n  the  expression  given  by  equation 
(66)  retaining  only  the  first  two  terms,  we  obtain  the  equa- 
tion 


This  equation  contains  the  three  unknowns  b,  €2,  and  </>r 
They  may  be  determined  by  applying  the  equation  at  con- 
stant temperature  to  a  substance  at  three  densities  not 
differing  much  from  each  other.  The  values  of  n  and  Vt 
may  then  at  once  be  obtained  similarly  as  in  Section  29. 

It  may  be  noted,  however,  that  it  is  preferable  to  deter- 
mine the  quantities  6,  €2,  and  <£„  if  possible,  without  using 
simultaneous  equations,  or  using  as  few  as  possible,  as  this 
gives  more  reliable  results.  For  the  variables  of  a  set  of 
simultaneous  equations  may  usually  be  varied  over  a 
considerable  range  and  yet  approximately  satisfy  the  equa- 
tions, and  hence  slight  errors  in  the  constants  of  the  equa- 
tions (furnished  by  experiment)  may  considerably  affect 
the  values  obtained  for  the  variables. 

(g)  In  Sub-section  (a)  of  this  Section  it  was  shown  that 
the  values  of  T?  for  substances  in  the  gaseous  state  obey 
the  relation  of  corresponding  states.  This  is  also  found  to 
hold  when  the  substances  are  not  in  the  gaseous  state,* 
as  is  shown  by  the  approximate  constancy  of  the  ratio  of 
171  to  772  corresponding  to  the  temperatures  Tc/2  and  47V7, 
shown  by  Table  XVII  for  a  number  of  liquids.  Since  the 
quantities  P^+p,  v,  T,  and  K,  in  equation  (92)  obey  this 
*  R.  D.  Kleeman,  Proc.  Camb.  Phil  Soc.,  Vol.  XVI,  Ft.  7,  p.  633. 


PROPERTIES  OF  THE  INTERFERENCE  FUNCTION  141 
TABLE  XVII 


ETHELENE-BROMIDE  (C4H4Br2) 

ETHELENE  CHLORIDE  (C4H4C12) 

*°c. 

ni  and  1)2. 

«/*». 

<°c. 

171  and  »/2. 

r)i/r)2. 

18.4 
60 

.01767 
.009985 

1.770 

7.7 

47.8 

.01000 
.005903 

1.693 

ETHYL  PROPIONATE  (CoHi0O2) 

BUTYRIC  ACID  (C4H8O2) 

0 

38.6 

.007040 
.003864 

1.822 

32.5 
76.1 

.01255 
.007016 

1.777 

THIOPHEN  (C4H4S) 

BENZENE  (C6H6) 

22.1 
64.3 

.00644 
.00408 

1.579 

3.8 
43.3 

.008680 
.004740 

1.831 

ETHYL  BENZENE  (C8Hi0) 

PROPIONIC  ACID  (C3H6O2) 

36.7 
80.9 

.00550 
.00357 

1.540 

33.4 

77.2 

.009169 
.005605 

1.635 

OCTANE  (C8Hi8) 

CARBON  TETRACHLORIDE  (CC14) 

11.6 
52.2 

.006060 
.003783 

1.603 

5 
44.7 

.0124 
.00696 

1.781 

relation  (Section  28  and  Sub-section  (a)  of  this  Section), 
it  follows  from  this  equation  that  the  factor  1 +<£>,,  also 
approximately  obeys  the  relation  of  corresponding  states. 

We  may  therefore  write  l+^  —  ^[~^j  — ),  where  Tc  denotes 

\  *     PI 

the  critical  temperature,  and  pc  the  critical  density  of  the 
substance. 


142  THE  NATURE  OF  MOLECULAR  MOTION 

35.  Formula  for  the  Viscosity  of  Mixtures. 

It  will  be  evident  from  an  examination  of  the  investiga- 
tion in  the  previous  Section  that  the  effect  of  each  molecule 
of  a  substance  on  its  viscosity  is  additive  in  character. 
Therefore  in  the  case  of  a  mixture  the  effect  of  the  different 
sets  of  molecules  is  additive.  It  will  easily  be  seen  there- 
fore that  in  the  case  of  a  mixture  of  molecules  e  and  r  the 
viscosity  is  given  by 

.....        (98) 


where  l^r  denotes  the  mean  momentum  transfer  distance 
of  a  molecule  r  of  absolute  mass  mar,  nr  the  number  of  mole- 
cules r  crossing  a  square  cm.  from  one  side  to  the  other  per 
second,  and  the  remaining  symbols  have  similar  meanings 
with  respect  to  the  molecules  e.  For  lnT  and  l,,e  we  may  write 


),     ....     (99) 

Vr       (/  r-LVer 

and 

' 


similarly  as  in  the  previous  Section,  where  vr  denotes  the 
volume  of  the  mixture  containing  a  gram  molecule  of  mole- 
cules r,  b'T  the  apparent  volume  of  the  molecules  r  and  e 
in  the  volume  vr  with  respect  to  the  motion  of  a  molecule  r 
(Section  20),  K\T  a  characteristic  constant  of  the  molecules 
r  when  the  mixture  is  in  the  gaseous  state,  Ner  the  total 
concentration  of  the  molecules  e  and  r,  N  the  number  of 
molecules  in  a  gram  molecule  of  a  pure  substance,  ^>,r  the 
effect  of  the  molecules  of  the  mixture  on  a  molecule  r  inter- 
acting with  a  molecule  r  or  e,  or  the  interference  function 
of  the  mixture  with  respect  to  the  molecules  r,  and  the 
remaining  quantities  have  similar  meanings  with  respect  to 


VISCOSITY  EQUATION  FOR  GASEOUS  MIXTURES     143 

the  molecules  e.  It  will  be  recognized  that  in  deducing  the 
foregoing  two  equations  along  similar  lines  as  equation 
(88),  we  have  written 

N  N 

l'v=K'v  inrCl+S,,,  and  Z'ne  =*'„«—  !+<*>„,), 

-iVer  IV  er 

or  replaced  volume  by  concentration,  which  in  the  case 
of  mixtures  is  more  convenient.  It  should  be  noted  that  in 

N 
the  case  of  a  pure  substance  —  say  e,  we  have  v=  —  .      It 

is  also  more  convenient  to  regard  <!>'„«,  and  <$',,,.  as  a  series 
of  powers  of  the  concentration  instead  of  the  volume.  As 
a  first  approximation  we  may  write 


and 


similarly  as  in  the  previous  Section.  It  will  be  easy  to 
see  that  the  various  quantities  involved  are  functions  of  the 
ratio  £  of  the  number  of  molecules  r  to  e,  as  well  as  of  their 
nature. 

On  applying  equation    (98)  to  the  gaseous  state  after 
substituting  from  equations  (99)  and  (100),  it  becomes 


Vv-M.,          (101) 


by  the  help  of  equation  (21),  and  since 

ve  N  .N     AT 

£  =  -,     and     —  +—  =  Ncr, 

Vr  Ve        Vr 

where    N  denotes    the    number  of    molecules    in    a  gram 
molecule,    and    mr   and    me   the    molecular    weights    of    a 


144       THE  NATURE  OF  THE  MOLECULAR  MOTION 

molecule  r  and  e  respectively  in  terms  of  that  of  the  hydro- 
gen atom.  This  equation  expresses  a  relation  between  the 
quantities  K.\T  and  K'^,  which  is  of  use  in  determining  them. 
Approximate  values  of  K'^  and  K\C  may  be  obtained 
from  the  values  of  K^  and  K^  corresponding  to  the  con- 
stituents separated  from  each  other  and  in  the  gaseous  state. 
In  the  case  of  a  pure  substance  r  in  the  gaseous  state 

N 
I,,,  =  a,,,.  vr  =  K,f  — .    Thus  l^  varies  inversely  as  the  probability 

of  a  molecule  r  coming  under  the  influence  of  another  mole- 
cule r  under  given  conditions,  which  is  proportional  to  Nr, 
and  hence  l/K,r  is  the  probability  factor  of  Nr.  In  the  case 
of  a  mixture  of  molecules  r  and  e  in  the  gaseous  state 
the  transfer  distance  l^r  varies  inversely  as  the  proba- 
bility of  a  molecule  r  coming  under  the  influence  of  a  mole- 
cule r  or  e.  This  probability  consists  of  the  sum  of  the 
probabilities  of  a  molecule  r  coming  under  the  influence 
of  another  molecule  r,  and  of  coming  under  the  influence 
of  a  molecule  e,  since  these  processes  are  independent. 

N    1 
Now  the  first  probability  is  equal  to  •—  — ,  where  K,r  refers 

J\      Kyr 

to  the  molecules  r  in  the  pure  state,  since  the  two  proba- 
bilities are  independent  and  we  may  therefore  suppose  the 
molecules  e  absent.  The  second  probability  is  equal  to 

AT  1  1 

-^  — ,  where  —  is  the  probability  factor  in  this  case.  Hence 
N  KX  KX 

we  have 


Ner 

:  AT 


or 


_1_        fA^        ATeM 


, 

(102) 


CHARACTERISTIC  FUNCTION  APPROXIMATION     145 

The  factor  I/KX  may  be  expressed  as  a  mean  of  the  factors 
I/K^  and  I/K^  referring  to  the  substances  r  and  e  in  the 
pure  state.  Thus  we  may  write 

or  we  may  write 


1=1)1+11 

KX       2\Kr,r       Kr,e\' 


which  would  hold  approximately.  Thus  an  approximate 
value  of  K'^  could  be  obtained  from  equation  (102)  in  terms 
of  quantities  referring  to  the  constituents  of  the  gaseous 
mixture  in  the  pure  state.  Similarly  an  approximate  expres- 
sion for  K'^  may  be  obtained.  The  accuracy  of  the  values 
obtained  may  be  checked  by  means  of  equation  (101). 

Approximate  values  of  $'  v  and  0',^  may  be  calculated 
from  the  values  of  0nr  and  </>,,<,  referring  to  the  isolated 
constituents  of  the  mixture.  In  the  case  of  a  pure  sub- 
stance r  the  term  Q^N^/N2  is  a  measure  of  the  probability 
of  a  molecule  r  coming  under  the  influence  of  another  mole- 
cule r  and  the  pair  coming  under  the  influence  of  a  third 
molecule  r.  In  the  case  of  a  mixture  of  molecules  r  and  e 
the  term  4>'wN2er/N21  is  correspondingly  a  measure  of  the 
probability  of  a  molecule  r  coming  under  the  influence  of 
another  molecule  r  and  the  pair  coming  under  the  influence 
of  a  molecule  r  or  e,  or  a  molecule  r  coming  under  the  influ- 
ence of  a  molecule  e  and  the  pair  coming  under  the  influence 
of  a  molecule  r  or  e.  These  four  probabilities  may  be  taken 
as  approximately  independent  of  each  other,  and  the  total 
probability  therefore  equal  to  their  sum.  We  may  there- 
fore write 


,          .          .         . 

^X~  "  ~  ~*  ' 


146          THE  NATURE  OF  MOLECULAR  MOTION 

where  <t>nr  refers  to  molecules  r  in  the  pure  state,  and  $x, 
<f>v,  and  </>2,  are  appropriate  probability  factors.  These 
factors  may  approximately  be  expressed  in  terms  of  quan- 
tities referring  to  the  molecules  r  and  e  in  the  pure  state. 
Thus  we  may  write 


approximately;  or  we  may  write 


approximately.     Similarly  an   approximate   expression  for 
d,  \e  may  be  obtained. 

The  quantities  just  considered  may  be  determined  di- 
rectly by  the  following  method.  In  conformity  with 
equation  (102)  we  may  write 


v 
and 


On  substituting  for  K'^  and  «^e  from  these  equations  in 
equation  (101)  and  applying  it  to  a  gaseous  mixture  at 
three  different  relative  concentrations  at  constant  tem- 
perature three  simultaneous  equations  are  obtained  from 
which  the  constants  ar,  are,  and  ae  may  be  determined. 


INTERFERENCE  FUNCTION  APPROXIMATION       147 

Similarly  in    conformity  with  equation  (103)  we  may 
write 


and 


On  substituting  for  4)\r  and  tf^  in  equation  (98)  transformed 
by  means  of  subsequent  equations  as  indicated,  and  apply- 
ing it  to  four  different  relative  concentrations  of  the  mixture 
at  constant  temperature  four  simultaneous  equations  will 
be  obtained  from  which  the  constants  br,  brre,  bree,  and  be, 
may  be  determined.  The  values  of  b'T  and  b'e  involved, 
it  may  be  pointed  out,  may  be  obtained  by  the  help  of 
Section  29. 

Knowing  the  values  of  the  foregoing  seven  constants 
and  the  values  of  b'e  and  b'r,  the  viscosity  may  be  calcu- 
lated for  any  density  and  relative  concentration  of  the 
constituents. 

Similarly  a  mixture  of  more  constituents  than  two  may 
be  treated. 

36.  The  Coefficient  of  Conduction  of  Heat. 

When  heat  flows  from  one  part  of  a  substance  to  another 
at  a  different  temperature  without  a  bodily  transference 
of  matter  taking  place  the  heat  is  said  to  be  propagated 
by  conduction.  The  heat  energy  is  then  transferred  from  one 
molecule  to  another  in  the  direction  of  the  flow  of  heat 
through  their  interaction  by  means  of  their  forces  of  attrac- 
tion and  repulsion.  The  coefficient  of  conduction  of  heat  is 
usually  denned  in  connection  with  the  flow  of  heat  across 
a  slab  of  material  of  thickness  d  and  infinite  extent  whose 
surfaces  are  kept  at  the  different  temperatures  ti  and  fa, 


148  THE  NATURE  OF  MOLECULAR  MOTION 

where  ti>t2  say.     The  quantity  of  heat  Q  transferred  per 
second  per  square  cm.  of  each  surface  is  defined  by 


(104) 


where  C  is  a  constant  which  is  called  the  coefficient  of  con- 

duction of  heat,  and  -W-^  is  called  the  temperature  gradient 
d 

of  the  flow  of  heat.  When  the  gradient  is  equal  to  unity 
it  follows  from  the  equation  that  the  coefficient  C  is  equal 
to  the  quantity  of  heat  transferred  from  one  side  of  the 
slab  to  the  othe.r  per  second  per  square  crn.  of  surface. 
The  lines  of  flow  of  heat  in  the  foregoing  arrangement 
would  obviously  be  everywhere  perpendicular  to  each  sur- 
face of  the  slab.  In  practice  with  a  slab  of  finite  dimensions 
this  is  realized  only  near  the  central  portion,  and  the  amount 
of  heat  transferred  is  therefore  measured  for  this  portion 
only,  the  rest  of  the  slab  acting  as  a  guard  ring  arrangement. 
As  in  the  case  of  viscosity,  the  flow  of  heat  in  a  substance 
is  directly  connected  with  the  nature  of  the  motion  of  a 
molecule,  and  it  will  therefore  be  necessary  to  define  a 
quantity  connected  with  the  path  described  by  a  molecule 
in  migrating  from  one  portion  of  the  substance  to  another. 

37.  The   Mean   Heat    Transfer   Distance   of  a 
Molecule  in  a  Substance. 

Consider  a  substance  which  is  at  a  higher  temperature 
in  the  plane  AB,  Fig.  14,  than  in  the  parallel  plane  CD, 
and  which  has  a  uniform  temperature  gradient  between 
the  planes.  Let  abc  denote  the  path  of  a  molecule  migrat- 
ing progressively  into  layers  at  lower  temperatures.  The 
molecule  loses  heat  energy  in  continually  varying  amounts 
to  the  medium  at  the  expense  of  its  kinetic  energy  (Section 
16),  and  the  change  in  its  potential  energy  of  attraction 


THE  TRANSFERENCE  OF  HEAT  BY  A  MOLECULE  149 


brought  about  by  passing  progressively  into  denser  layers 
of  the  substance.  We  may  suppose  that  the  medium  acquires 
and  loses  energy  at  certain  points  only  near  the  path  of  the 
migrating  molecule,  which  energy  is  abstracted  from,  or 
transferred  to,  the  migrating  molecule  as  the  case  may  be. 
The  amount  of  energy  lost  at  a  point  will  be  taken  equal 
to  772>Sm,  where  T^  denotes  the  absolute  temperature  at  the 
point,  and  Sm  the  internal  specific  heat  at  constant  pressure 
of  the  molecule,  while  the  energy  acquired  at  the  point  will 
be  taken  equal  to  T\Sm,  where  T\  denotes  the  absolute 
temperature  at  the  preceding  point.  The  internal  specific 
heat  Sm  of  a  molecule  is  equal  to  the  internal  specific  heat 


FIG.  14. 

at  constant  pressure  of  a  gram  of  molecules  (Section  13) 
divided  by  the  number  of  molecules  it  contains.  The  line 
joining  two  consecutive  points  will  be  called  a  heat  transfer 
distance  in  the  substance,  since  the  molecule  abstracts  the 
quantity  of  heat  TiSm  at  one  extremity  of  the  distance  and 
transfers  it  to  the  other  extremity,  while  at  the  latter  point 
it  abstracts  the  quantity  of  heat  T2Sm  and  transfers  it  to 
the  other  extremity  of  the  adjacent  distance,  etc.  Let  us 
suppose,  similarly  as  in  the  case  of  viscosity,  that  the  posi- 
tions of  the  points  associated  with  the  path  of  a  migrating 
molecule  are  so  selected  that  the  number  of  heat  transfer 


150  THE  NATURE  OF  MOLECULAR  MOTION 

distances,  and  the  number  of  molecular  paths,  cutting  a 
plane  of  one  square  cm.,  are  equal  to  one  another,  and  that 
all  directions  of  a  transfer  distance  in  space  are  equally 
probable.  These  two  conditions  determine  the  positions 
and  lengths  of  the  transfer  distances.  These  distances  are 
not  equal  to  each  other,  but  are  grouped  about  a  mean 
distance  lc  according  to  the  distribution  law  of  Clausius 
given  in  Section  31. 

The  mean  heat  transfer  distance  of  a  molecule  is  evi- 
dently not  equal  to  the  mean  momentum  transfer  distance 
in  viscosity,  for  one  refers  to  the  transfer  of  momentum 
and  the  other  to  energy  under  different  conditions.  Experi- 
mental evidence  will  be  considered  later  showing  that  that 
is  so.  In  treatises  on  the  Kinetic  Theory  of  Gases  both  dis- 
tances are  usually  assumed  to  be  equal  to  the  average  dis- 
tance between  two  consecutive  collisions  of  a  molecule. 

We  will  now  deduce  formulae  involving  the  mean  heat 
transfer  distance  as  defined  in  this  Section. 

38.  Formulae  for  the  Coefficient  of  Conduction 
of  Heat  in  Terms  of  Other  Quantities. 

Let  us  suppose  that  the  temperature  of  a  substance  is 
higher  in  the  plane  AB,  Fig.  15,  than  in  the  parallel  plane 
CD.  Suppose  that  n  molecules  cross  the  plane  EF  from 
one  side  to  the  other.  This  migration  of  molecules  gives 
rise  to  a  transference  of  heat  across  the  plane  from  top  to 
bottom  which  per  square  cm.  is  numerically  equal  to  the 
coefficient  of  conduction  of  heat  if  the  temperature  gradient 
is  unity.  This  amount  of  heat  may  be  expressed  in  terms 
of  other  quantities  as  follows.  Consider  the  heat  transfer 
distances  which  cut  the  plane  EF  and  are  inclined  at  an 
angle  6  to  a  perpendicular  to  the  plane.  For  purposes  of 
calculation  we  may  suppose  that  the  upper  extremities  of 
these  distances  are  located  at  the  same  point  which  may  be 


THE  TRANSFERENCE  OF  HEAT  ACROSS  A  PLANE      151 

taken  to  lie  in  the  plane  EF.  It  follows  then  from  the  figure, 
which  shows  a  section  of  a  hemisphere  of  radius  z  having  the 
point  mentioned  as  center,  that  the  number  of  these  dis- 
tances is  equal  to 


l~~" 


where  z  denotes  the  length   of  one  of  these  distances,  r 
z  sin  6,  and  the  number  is  thus  equal  to  n  sin  6  •  dB. 


FIG.  15. 

If  n^  denote  the  number  of  the  f oregoinj  distances  whose 
lengths  lie  between  z  and  z+dz,  we  have 


nz  =  n  sin  6  -  d0  •  j-^ 


z  -L 


dz, 


by  the  help  of  equation  (81).  Each  of  the  molecules  cor- 
responding to  the  foregoing  distances  abstract  the  energy 
T\Sm  from  the  plane  EF  which  is  at  the  temperature  TI, 
and  transfers  it  to  the  plane  GH  which  is  at  the  temperature 
TV  An  equal  number  of  molecules  inclined  at  the  angle 
6  to  a  line  at  right  angles  to  the  plane  EF  move  in  the  oppo- 
site direction.  Each  of  these  molecules  abstracts  the  energy 
T2Sm  from  the  plane  GH  and  transfers  it  to  the  plane  EF. 
Thus  on  the  whole  these  two  sets  of  molecules  transfer  the 


152  THE  NATURE  OF  MOLECULAR  MOTION 

energy  nz(TiSm—T2Sm)  across  the  plane  EF.     On  taking 
the  temperature  gradient  equal  to  unity  we  have 

- 

' 


z  cos  e 

and  the  preceding  expression  for  the  energy  transfer  becomes 
nz  Sm  z  cos  6. 

The  total  energy  transferred  per  square  cm.  across  the 
plane  EF  is  obtained  on  substituting  for  nz  in  the  foregoing 
expression  and  integrating  it  from  0  to  oo  with  respect  to 
the  transfer  distance  z,  and  from  0  to  7r/2  with  respect  to 
the  angle  6,  which  gives 


fir- 

=  nSm 

JQ  Jo 


Z2       lc 

cos  0sin  0,-^e      -dd-dz, 


where  C  denotes  the  coefficient  of  conduction  of  heat.  The 
integral  in  this  equation  can  be  shown  to  be  equal  to  lc 
(Section  34),  which  reduces  the  equation  to 


(105) 


where  n  denotes  the  number  of  molecules  crossing  a  square 
cm.  from  one  side  to  the  other,  lc  the  mean  heat  transfer 
distance,  and  Sm  the  internal  specific  heat  at  constant  pres- 
sure per  molecule.  If  Sff  denote  the  internal  specific  heat 
per  gram  at  constant  pressure  of  the  substance  we  have 

Sm=Sgma,    ......     (106) 

where  ma  denotes  the  absolute  molecular  weight  of  a  mole- 
cule, and  the  preceding  equation  may  be  written 

C  =  nmaSalc  .......     (107) 

By  means  of  this  equation  the  value  of  lc  for  any  state  of 


AN  INFERIOR  LIMIT  OF  THE  FREE  PATH          153 

matter  may  be  calculated,  since  C  and  Sg  may  be  measured 
directly  and  n  determined  indirectly  according  to  Section  29. 
The  quantity  lc  may  be  expressed  in  terms  of  other 
quantities  similarly  as  the  quantity  Z,  in  Section  34,  by 
representing  the  substance  by  another  substance  possessing 
the  same  coefficient  of  conduction  of  heat,  internal  specific 
heat  at  constant  pressure,  and  expansion  pressure,  but 
whose  molecules  possess  no  apparent  volume  6.  The  coeffi- 
cient of  conduction  of  the  representative  substance  is  evi- 
dently given  by 

C  =  n'maSffl'c, 

where  n'and  l'c  referring  to  the  representative  substance 
have  meanings  similar  to  n  and  lc  in  equation  (107).  Since 
the  representative  substance  has  the  same  expansion  pres- 
sure as  the  original  substance,  and  the  molecules  of  the 
former  substance  have  no  volume,  we  have  according  to 
equations  (38)  and  (45)  that 


v-b      A/2' 

where  the  quantities  n,  6,  Pn,  p,  v,  and  A  refer  to  the  original 
substance.  The  preceding  equation  may  therefore  be  writ- 
ten 


C=  -jnmaSil'c  =       "ma£/c.    .    '.     (108) 
v  —  o  A. 

This  equation  determines  l'c  since  each  of  the  other  quan- 
tities it  contains  may  be  determined  from  the  original 
substance. 

The  relation  between  lc  and  l'c  according  to  equations 
(107)  and  (108)  is  given  by 


(109) 


and  thus  V  c  is  an  inferior  limit  of  lc. 


154  THE  NATURE  OF  MOLECULAR  MOTION 

Equation  (108)  may  be  given  another  form  along  the 
same  lines  as  equation  (84)  in  Section  34.  If  the  mole- 
cules in  the  representative  substance  were  devoid  of  molec- 
ular forces  which  extend  beyond  a  small  distance  from  each 
molecule,  the  quantity  lc  would  vary  inversely  as  the  chance 
of  one  molecule  meeting  another,  or  be  proportional  to  the 
volume  v,  and  we  may  write 

l'c=KcV, 

where  KC  is  a  constant  at  constant  temperature.  The  exist- 
ence of  molecular  forces  extending  some  distance  from  each 
molecule  gives  rise  to  the  interaction  of  two  molecules  not 
being  independent  of  the  surrounding  molecules,  and  hence 
l'c  not  being  proportional  to  v.  This  effect  may  be  expressed 
by  a  factor  introduced  into  the  right-hand  side  of  the  fore- 
going equation,  which  is  obtained  in  the  same  way  as  a 
similar  factor  in  Section  34.  Since  the  chance  of  a  mole- 
cule interacting  with  another  molecule  and  the  pair  coming 
under  the  influence  of  r  molecules,  is  proportional  to  1/V+1, 
the  factor  in  question  may  be  written 


The  quantities  <£'c,  4>"c,  .  .  .  ,  which  are  functions  of  T 
and  v,  insensitive  to  v,  are  not  identical,  it  should  be  noted, 
with  the  quantities  <£',,,  </>"„  .  .  .  ,  in  Section  34,  because 
the  quantities  /,  and  lc  have  somewhat  different  meanings, 
but  they  will  probably  not  differ  much  from  each  other. 
The  quantities  lc  and  Vc  are  therefore  given  by  the  equations 


and 

l'c=Kcv(l+3>c)  ........     (112) 


VARIOUS  FORMULA  FOR  HEAT  CONDUCTION      155 

Equation  (111)  may  be  interpreted  similarly  as  equa- 
tion (88).  The  interaction  of  two  molecules  is  modified 
by  the  surrounding  molecules,  an  effect  which  was  called 
molecular  interference.  The  value  of  the  apparent  molec- 
ular volume  b  in  the  equation  depends  to  some  extent 
(which  is  probably  small)  on  molecular  interference,  while 
$c  depends  wholly  on  it.  The  latter  quantity  may  therefore 
be  called  the  interference  function  in  heat  conduction. 
In  value  it  probably  differs  not  much  from  that  of  the 
interference  function  $,,  in  viscosity.  Therefore  according  to 
equation  (111)  the  effect  of  molecular  interference  on  lc  which 
is  not  represented  by  b  (a  quantity  which  also  occurs  in 
equation  (88)),  is  represented  by  the  function  3>c,  which 
may  now  be  used  without  referring  to  the  representative 
substance. 

Equation  (107)  may  therefore  be  written  in  the  forms 


(113) 


I,  ....     (114) 

o\v  —  uj  I 

and 

'<),  (115) 


by  the  help  of  equations  (35)  and  (46),  and  remembering 
that  Ncmav  =  m.  The  values  of  n,  b,  and  Vt,  the  number  of 
molecules  crossing  a  square  cm.,  the  apparent  molecular 
volume  in  cubic  cms.  per  gram  molecule,  and  the  total 
average  velocity  in  cms.  respectively,  may  be  determined 
by  the  method  described  in  Section  29.  The  value  of  the 
intrinsic  pressure  P»,  which  must  be  expressed  in  dynes 
similarly  as  the  pressure  p,  may  be  determined  by  the 
method  described  in  Section  21.  The  internal  specific 
heat  Sg  per  gram  at  constant  pressure,  and  the  conduction 


156  THE  NATURE  OF  MOLECULAR  MOTION 

of  heat  C  per  second  across  a  square  cm.  at  right  angles  to 
unit  heat  gradient,  must  be  expressed  in  terms  of  the  same 
heat  units,  which  for  convenience  may  be  taken  the  calorie, 
or  the  mechanical  heat  unit  the  erg.  The  calculated  values 
of  n,  6,  and  Vt  depend  on  the  distribution  of  velocities  as- 
sumed for  the  molecules;  they  may  be  corrected  according 
to  Maxwell's  law,  if  desired,  in  the  way  described.  The 
value  of  A  in  equation  (115)  not  corrected  for  the  distri- 
bution of  molecular  velocities  is  given  by 

A  =5.087X10  ~20  VTmt 
according  to  Section  11.    If  corrected  according  to  Maxwell's 

/o~ 

law  the  value  has  to  be  multiplied  by  \--,  or  1.085.     It 


will  appear,  however,  from  Sub-section  a  that  the  calculated 
value  of  KC  similarly  corrected  involves  the  factor  1.085  and 
that  therefore  the  uncorrected  values  of  A  and  KC  may  be 
used  in  equation  (115).  Since  p  is  independent  of  the 
molecular  distribution  of  velocities  (Section  8),  and  Pn  is 
obviously  so  by  nature,  the  dependence  of  the  factor  KC/A 
in  equation  (115)  on  this  distribution  is  represented  in 
equations  (114)  and  (113)  by  factors  involving  the  quan- 
tities n,  Kc,  b,  and  Vt.  The  uncorrected  values  of  these 
quantities  may  therefore  be  used  in  these  equations.  It 
may  be  noted  here  that  the  quantity,  Sg  is  probably  very 
approximately,  if  not  altogether,  independent  of  the  dis- 
tribution of  molecular  velocities.  This  follows  according 
to  Section  13  from  the  fact  that  it  depends  mainly  on  the 
change  in  kinetic  energy  and  change  in  potential  energy  of 
attraction  of  a  gram  of  substance  during  a  degree  change  of 
temperature,  both  changes  being  independent  of  the  dis- 
tribution of  molecular  velocities. 

The  value  of  KC  in  these  equations  for  a  given  substance 
is  determined  by  applying  equation  (113)  to  the  substance 


THE  HEAT  CONDUCTION  EQUATION  FOR  A  GAS     157 

in  the  perfectly  gaseous  state,  which  corresponds  to  v=  oo  , 
in  which  case  the  equation  becomes 


(116) 


by  means  of  equation  (21),  where  Sa  is  now  equal  to  the 
specific  heat  at  constant  volume.  The  quantity  KC,  it  should 
be  noted,  is  not  identical  with  the  quantity  Kn.  It  is  a  char- 
acteristic constant  depending  on  the  molecular  forces  and 
volume  of  the  substance. 

The  values  of  the  quantities  n,  Vt,  6,  Pn,  p,  and  Se  are 
affected  to  a  certain  extent  by  the  influence  of  the  mole- 
cules of  the  substance  on  two  molecules  while  interacting, 
or  influenced  by  molecular  interference.  The  quantity  3>c 
therefore  represents  the  portion  of  a  similar  effect  on  the 
heat  conductivity  of  a  substance  which  is  not  represented 
by  the  foregoing  quantities  in  equations  (113),  (114),  and 
(115).  These  equations  may  now  be  used  without  referring 
to  the  representative  substance. 

It  follows  similarly  as  in  Section  34  in  connection  with 
the  quantity  <$,,,  that  the  terms  of  the  series  for  3>c  decrease 
rapidly  in  magnitude.  As  a  first  approximation  we  may 
therefore  retain  only  the  term  <f>'c/v2,  or  replace  the  series 
by  the  term  <f>c/vx,  where  x  would  differ  little  from  2.  It 
will  readily  be  recognized  that  the  quantities  <f>'c,  <f>"e,  .  .  .  , 
in  the  series  for  <£c  are  functions  of  T  and  v,  which  are  not 
very  sensitive  to  v. 

From  equations  (107)  and  (83)  we  have 


which  gives  the  ratio  of  the  mean  heat  transfer  distance 
associated  with  a  molecule  of  a  substance  to  the  mean 
momentum  transfer  distance.  This  would  furnish  some 


158  THE  NATURE  OF  MOLECULAR  MOTION 

information  about  the  difference  in  the  effect  of  molecular 
interference  on  the  chances  of  a  migrating  molecule  losing 
momentum  or  kinetic  energy  to  a  medium  under  the  con- 
ditions of  viscosity  and  heat  conduction  respectively,  since 
these  chances  are  respectively  proportional  to  I//,  and 
l/lc. 

On   substituting   for   lc  and  lv  in  the  foregoing  equation 
from  equations   (111)  and (88),  it  becomes  after  rearranging 

=  ^4  (118) 


The  quantities  Kn  and  KC  in  this  equation  may  be  determined 
from  the  viscosity  and  heat  conduction  of  the  substance  in 
the  gaseous  state,  while  the  quantities  C,  Sg,  and  77  (which 
refer  to  any  state)  may  be  determined  directly.  The  equa- 
tion may  thus  be  used  to  compare  the  quantities  3>c  and 
<f>r  Since  KC  and  K^  do  not  depend  on  molecular  interference 
according  to  their  definitions,  and  Sg  also  does  not  from 
its  nature,  while  both  3>c  and  <£,  depend  entirely  on  it,  it 
follows  from  equation  (118)  that  an  increase  of  <I>C  rela- 
tive to  <!>,,  is  attended  by  an  increase  of  C  relative  to 
77. '  Thus  this  equation  gives  some  information  on  the  relative 
effects  of  molecular  interference  on  the  heat  conductivity 
and  viscosity  of  a  substance. 

A  few  applications  of  the  foregoing  investigation  will 
now  be  given. 

(a)  Equation  (116)  gives  the  coefficient  of  heat  conduc- 
tion of  a  gas.  Since  KC  is  independent  of  the  density  of  the 
gas,  it  follows  from  the  equation  that  this  also  holds  for  the 
quantity  C.  Experiment  has  demonstrated  the  truth  of  this 
result.  It  would  of  course  cease  to  hold  when  the  dimen- 
sions of  the  vessel  in  which  conduction  takes  place  are 
comparable  with  the  length  of  the  mean  heat  transfer  dis- 
tance, according  to  the  conditions  on  which  the  deduction 
of  the  equation  is  based. 


THE  EFFECT  OF  TEMPERATURE  ON  CONDUCTION    159 

The  dynamical  mechanism  underlying  this  result  may 
be  illustrated  by  the  following  considerations.  A  molecule 
in  a  gas  moving  in  the  direction  of  the  flow  of  heat  transfers 
a  certain  amount  of  heat  to  the  medium  at  the  end  of  each 
transfer  distance.  If  the  concentration  of  the  gas  is  halved 
the  length  of  each  transfer  distance  is  doubled,  while  the 
heat  transferred  at  the  end  of  each  transfer  distance  is  also 
doubled  since  the  temperature  gradient  remains  the  same. 
Since  a  change  in  molecular  concentration  of  a  gas  does  not 
alter  the  molecular  velocities,  the  heat  transferred  per  second 
by  a  molecule  moving  between  two  walls  at  different  tem- 
peratures in  the  latter  case  is  double  that  in  the  former. 
But  since  the  number  of  molecules  per  cubic  cm.  available 
for  heat  transference  in  the  latter  case  is  half  the  number 
available  in  the  former  case,  the  total  heat  transferred  is  the 
same  in  each  case. 

The  value  of  KC  is  found  to  increase  with  increase  of  tem- 
perature, as  is  the  case  with  the  similar  quantity  K,,  con- 
nected with  the  viscosity  of  the  substance.  This  is  shown 
for  a  few  gases  by  Table  XVIII.  The  reason  is  undoubtedly 
the  same  as  that  holding  in  the  case  of  viscosity,  namely 
that  the  greater  the  velocity  of  a  molecule  the  shorter  the 
time  it  is  under  the  influence  of  another  molecule,  and  the 
smaller  its  chance  of  transferring  heat  energy,  this  chance 
being  measured  by  l/lc,  or  I/KC. 

The  values  of  KC  in  the  Table  are  not  corrected  for  the 
distribution  of  molecular  velocities  since  equation  (116) 
was  derived  from  equation  (113)  on  substituting  for  n 
the  expression  given  by  equation  (21).  If  they  are  cor- 
rected according  to  Maxwell's  law  each  value  has  to  be 

/Q— 

multiplied  by  -\-jj-,  or  1.085,  according  to  Section  11.     The 

same  remarks  apply  to  the  values  of  the  quantity  K,  in  the 
Table. 


160        •  THE  NATURE  OF  MOLECULAR  MOTION 
TABLE  XVIII 


C2H4.     TO  =  44 

i°C. 

Sg  cal.  per  gm. 

C103cal./cm.2  sec.         KC  10'° 

ic,  lO'o 

*</",, 

0 
100 

.3245 
.3403 

.0395               2.64 
.0506               2.77 

2.08 
2.38 

1.27 
1.16 

CO2.     TO  =  28 

0 
100 

.1541 
.1724 

.0307 
.0506 

3.45 
4.36 

2.42 
2.92 

1.43 
1.49 

H2.     TO  =  2 

-150 
0 

2.41 
2.41 

.1175 
.3270 

5.91 
11.04 

5.86 
6.69 

1.00 
1.65 

A.     TO  =39.9 

0 

.07379 

.03894 

9.61 

3.83 

2.51 

Hg.     TO  =  200.4 

203 

.01476 

.01846 

7.70 

1.78 

4.33 

On  applying  equations  (118)  and  (117)  to  the  gaseous 
state  it  follows  that 


since  ^,  =  0,  and  4>c=0,  under  these  conditions.     The  cal- 
culation of  the  ratio  KC/KV  for  different  gases  shows  that  it 


PROPERTIES  OF  THE  CHARACTERISTIC  FUNCTION     161 

is  not  equal  to  unity,  or  the  quantities  KC  and  K,,  are  not 
equal  to  each  other,  and  that  the  value  of  the  ratio  depends 
on  the  nature  of  the  gas  and  its  temperature,  as  is  shown 
by  Table  XVIII.  We  would  expect  that  these  quantities 
should  not  be  equal  to  each  other  since  they  do  not  mean 
exactly  the  same  thing.  For  1/7C,  or  l//cc,  is  a  measure  of 
the  chance  of  heat  energy  being  transferred  by  a  migrating 
molecule  to  its  medium  under  the  conditions  of  heat  conduc- 
tion, while  1/Z,,,  or  I/*,,  is  a  measure  of  the  chance  of  mo- 
mentum being  transferred  to  the  medium  under  the  con- 
ditions of  viscosity.  The  former  chance  is  smaller  than  the 
latter  in  gases,  since  according  to  the  Table  KC/K,,  or 

Y  is  greater  than  unity. 

tr, 

It  will  be  of  interest  to  give  a  more  definite  and  direct 
physical  significance  of  KC,  and  compare  it  with  that  obtained 
for  Kn  in  Sub-section  (a)  of  Section  34.  The  quantity  C 
represents  the  amount  of  energy  which  flows  per  second 
across  each  square  cm.  of  a  plane  in  a  substance  situated 
at  right  angles  to  a  unit  temperature  gradient.  Since  n 
molecules  cross  the  plane  per  square  cm.  per  second  in  each 
direction,  C/n  represents  the  energy  conveyed  by  a  mole- 
cule across  the  plane  on  crossing  it  and  recrossing  it  (some- 
time) in  the  opposite  direction.  According  to  equations 
(105)  and  (111)  this  energy  is  equal  to  KcSmv  in  the  case  of 
a  gas,  where  the  value  of  Sm  is  independent  of  the  pressure 
of  the  gas.  Thus  KC  represents  the  energy  conveyed  by 
a  molecule  per  unit  of  its  specific  heat  at  unit  volume  of  a 
gram  molecule  of  the  gas.  The  quantity  K,,,  we  have  seen, 
represents  the  momentum  conveyed  by  a  molecule  across  a 
plane  at  right  angles  to  unit  velocity  gradient  per  unit  mass 
of  the  molecule  at  unit  volume  of  a  gram  molecule  of  the  gas. 

It  can  be  shown  similarly  as  in  Section  34  that  a  mole- 
cule takes  a  time  proportional  to  v/V  in  crossing  and  re- 
crossing  a  plane.  The  amount  of  energy  conveyed  by  a 


162 


THE  NATURE  OF  MOLECULAR  MOTION 


molecule  per  second  across  a  plane  under  the  foregoing  con- 
ditions is  therefore  proportional  to  KC8mV,  and  thus  inde- 
pendent of  the  volume  of  the  gas. 

(6)  Equation    (117)    may    immediately    be    applied    to 
liquids  to  obtain  the  values  of  the  ratio  ljli\.    Table  XIX 

TABLE  XIX 


Substance. 

t°  C. 

Sg  cal.  per  gm. 

r> 

C  cal./cm.2  sec. 

yi, 

E.  bromide  .... 

15 

.2135 

.004212 

.03247 

.275 

E.  iodide  

15 

.1641 

.006231 

.03222 

.217 

Chloroform  .... 

15 

.237 

.006019 

.03288 

.202 

M.  alcohol.  .  .  . 

15 

.5868 

.006429 

.03495 

.131 

C.  tetrachloride 

0 

.2010 

.01351 

.032664 

.0981 

C.  disulphide  .  . 

15 

.242 

.003905 

.03343 

.363 

Benzene  

10 

.4066 

.007631 

.03333 

.107 

contains  the  values  for  a  number  of  substances.  It  will 
be  seen  that  they  depend  on  the  nature  of  the  substances, 
and  that  they  are  smaller  than  unity.  We  have  just  seen 
that  when  the  substances  are  in  the  gaseous  state  this  ratio 
is  greater  than  unity.  This  difference  is  due  to  the  effect 
of  the  attraction  of  the  molecules  of  a  substance  on  a  migrat- 
ing molecule  increasing  with  the  density  of  the  substance. 
It  indicates  that  an  increase  of  molecular  interference  in- 
creases to  a  greater  extent  the  chance  of  a  molecule  trans- 
ferring heat  energy  than  of  transferring  momentum  under 
the  conditions  of  heat  conduction  and  viscosity  respectively, 
since  these  chances  are  measured  by  l//c  and  1/Z,,. 

According  to  the  foregoing  result,  and  the  result  that 
KC/K,  is  greater  than  unity,  it  follows  from  equation  (118) 


and  (117)  that 


is  less  than  unity,  and  that  therefore 


<$,,  ><£<;,  or  the  interference  function  of  viscosity  is  greater 
than  the  interference  function  of  heat  conduction.  Since 
each  of  these  quantities  is  zero  for  matter  in  the  gaseous 


DETERMINATION  OF  INTERFERENCE  FUNCTION      163 

state,  an  increase  in  the  molecular  interference  as  brought 
about  by  an  increase  of  the  density  of  the  matter  already 
in  a  dense  state  would  therefore  increase  <£,,  to  a  greater 
extent  than  3>c.  Therefore  according  to  equation  (118) 
and  the  considerations  following  it  the  value  of  17  is  increased 
to  a  greater  extent  by  the  increase  of  molecular  interference 
brought  about  by  an  increase  01  density  of  the  matter  than 
the  value  of  C. 

(c)  Equation  (115)  may  be  used  to  obtain  the  value  of 
<£c  for  a  liquid,  and  hence  the  corresponding  value  of  <f>'c 
may  be  deduced.    The  value  of  Pn+p  required  for  the  cal- 
culations may  be  obtained  from  the  coefficients  of  expansion 
and   compression   of   the   liquid   according  to   Section   21. 
It  may  also  be  obtained  by  the  method  of  Section  46.    The 
value  of  KC  may  be  determined   by  means  of  equation  (116), 
which  applies  to  the  substance  in  the  gaseous  state.     At 
present  the  data  are  not  sufficient  to  permit  such  a  calcula- 
tion being  carried  out  for  a  substance. 

(d)  If  the  values  of  C  and  Sg  be  known  for  three  differ- 
ent densities  of  a  substance  not  differing  much  from  each 
other,  and  the  value  of  KC  be   calculated  by  means  of  equa- 
tion (116),  the  values  of  b  and  </>'c  may  be  determined  by 
means  of  a  modified   form  of  equation  (113)  obtained  on 
substituting  for  3>c  the  approximate  expression  <A'CA2,  and 
for  n  the  expression  given  by  equation  (66)  retaining  only 
the  first  two  terms.     The  equation  will  contain  the  three 
constants,  b,  €2,  and  <//c,  which  may  be  determined  by  apply- 
ing the  equation  to  the  substance  at  three  different  densi- 
ties.   This  operation  also  determines  incidentally  the  quan- 
tities n  and  Vt  similarly  as  in  Section  29. 

39.  Formulce  for  the  Coefficient  of  Conduction 
of  Pleat  of  a  Mixture  of  Substances. 

It  will  be  evident  from  a  consideration  of  the  investiga- 
tion in  the  previous  Section  that  in  the  case  of  a  mixture 


164  THE  NATURE  OF  MOLECULAR  MOTION 

the  heat  conductivity  is  equal  to  the  sum  of  the  effective 
conductivities  of  the  constituents.  Hence  in  the  case  of  a 
mixture  of  molecules  r  and  e  we  may  according  to  equation 
(105),  write  for  the  coefficient  of  conductivity 

C  =  nrSmrlcr+neSmelce,      ....     (120) 

where  Smr  and  Sme  denote  the  partial  molecular  specific 
heats  of  a  molecule  r  and  e  respectively  (Section  13),  lcr 
denotes  the  mean  heat  transfer  distance  of  a  molecule  r, 
Ice  that  of  a  molecule  e,  and  nr  and  ne  have  the  usual  meanings. 
Similarly  as  in  Section  35  we  may  write 


and 

t-  -V  ^r  (!+*'«),  .     .     .     .     (122) 

l/e      t/  e     IV  er 

where  the  symbols  have  similar  meanings,  that  is,  vr  denotes 
the  volume  of  the  mixture  containing  a  gram  molecule 
of  molecules  r,  N&  the  total  concentration  of  the  molecules 
e  and  rr,  b'r  the  apparent  molecular  volume  of  the  mixture 
which  appears  to  obstruct  the  motion  of  the  molecules  r, 
K'CT  the  characteristic  factor  of  the  transfer  path  of  a  mole- 
cule r  when  the  mixture  is  in  the  gaseous  state,  and  && 
the  heat  conductivity  interference  function  with  respect 
to  a  migrating  molecule  r,  while  the  other  symbols  have 
similar  meanings.  As  a  first  approximation  we  may  write 
as  before 


fy 


and 

*/       ^  ^ 


THE  CONDUCTIVITY  OF  GASEOUS  MIXTURES      165 

where  N  denotes  the  number  of  molecules  in  a  gram  mole- 
cule of  a  pure  substance,  and  <$>'  „  and  0'cc  are  functions  of 
T  and  Ner  but  which  are  insensitive  to  variations  in  Ner- 
On  substituting  from  equations  (122)  and  (121)  for  la 
and  lce  in  equation  (120)  and  applying  it  to  the  gaseous 
state,  it  becomes 


c=       '^+MS''  (123) 


by  the  help  of  equation  (21),  and  since 

Ve     N      N       ,,        „          Smr  ,    „         Sme 

£  =  -,    --  h  —  =  Ner,  bar  =  -  ,  and  Sft  =  -  , 

Vr      Ve        Vr  mar  Wlae 

where  Sgr  and  Sge  denote  the  partial  specific  heats  per  gram 
of  the  molecules  r  and  e  respectively,  £  the  ratio  of  the 
number  of  molecules  r  to  the  number  of  molecules  e  whose 
relative  molecular  weights  referred  to  the  hydrogen  atom 
are  mr  and  me,  and  N  denotes  the  number  of  molecules  in 
a  gram  molecule  of  a  pure  substance.  When  the  mixture 
is  in  the  perfectly  gaseous  state  the  values  of  Sgr  and  Sge  are 
the  same  as  for  the  pure  substances  in  the  gaseous  state  at 
constant  volume.  Equation  (123)  gives  a  relation  between 
the  quantities  *'„  and  *'«>  which  is  of  use  in  their  determina- 
tion. 

The  various  quantities  contained  in  the  foregoing  equa- 
tions are  evidently  functions  of  £,  the  ratio  of  the  constitu- 
ents. With  the  exception  of  the  quantities  vr,  ve,  b'e,  and 
6'r,  they  are  not  identical  with  the  quantities  contained  in 
the  equations  of  Section  35. 

The  values  of  K'  '„  and  /«.  may  be  approximately  calcu- 
lated from  the  values  of  KCT  and  Kce  referring  to  the  sub- 
stances isolated  and  in  the  perfectly  gaseous  state,  by  means 
of  a  formula  similar  to  equation  (102)  in  Section  35.  The 
values  thus  obtained  may  be  checked  by  means  of  equa- 


166          "THE  NATURE  OF  MOLECULAR  MOTION 

.tion  (123).  Similarly  the  values  of  <f>'cr  and  4>'Ce  may  approxi- 
mately be  calculated  by  a  formula  similar  to  equation 
(103).  The  foregoing  quantities  may  also  be  determined 
directly  by  the  method  given  in  the  Section  mentioned. 

(a)  It  is  instructive  to  apply  equation  (120)  to  the  con- 
duction of  heat  in  metals.  Since  the  molecules  of  a  metal 
are  more  or  less  in  relatively  rigid  positions  the  heat  energy 
must  in  the  main  be  transferred  by  carriers  which  are  able 
to  move  about  more  freely  than  the  molecules.  It  has 
been  supposed  that  these  carriers  consist  of  electrons  in  the 
free  state  which  arise  through  a  spontaneous  dissociation 
of  the  molecules  into  electrons  and  positively  charged  mole- 
cules. Equilibrium  exists  when  the  number  of  electrons 
produced  in  this  way  per  second  is  equal  to  the  number 
neutralized  through  combination  with  oppositely  charged 
parent  molecules.  The  state  of  equilibrium  may  therefore 
be  expressed  by  the  equation  of  mass-action. 


where  Nn  denotes  the  number  of  neutral  molecules  per 
cubic  cm.,  Ne  the  number  of  electrons  or  positively  charged 
molecules  per  cubic  cm.,  and  K  denotes  the  constant  of  mass- 
action.  This  constant  is  a  function  of  the  temperature, 
density,  and  nature,  of  the  metal. 

If  the  molecules  of  the  metal  are  designated  by  r  and 
the  electrons  by  e  in  equation  (120),  we  have  nr=0,  since 
the  molecules  are  more  or  less  in  relatively  fixed  positions, 
and  the  equation  becomes 

C  =  neSmelce         ......       (124) 

The  value  of  lce  is  probably  governed  by  the  spacing  of  the 
molecules  of  the  metal,  since  the  size  of  an  electron  is  very 
small  in  comparison  with  that  of  an  atom.  As  a  first  approxi- 
mation we  may  therefore  take  lce  equal  to  the  distance  of 


THE  MIGRATION  OF  ELECTRONS  IN  A  METAL      167 

separation  of  the  atoms  in  a  metal,  which  is  usually  of  the 
order  10  ~8  cm.  The  partial  specific  heat  Sme  of  an  electron 
probably  differs  little  from  \  ma  F2  =  2.012X10-1677  ergs, 
its  specific  heat  at  constant  volume  in  the  gaseous  state. 
Whatever  its  value,  it  cannot  be  greater  than  the  specific 
heat  per  atom  of  the  metal,  which  is  equal  to  twice  the 
specific  heat  it  would  have  in  the  gaseous  state.  We  may 
therefore  with  a  fair  degree  of  certainty  calculate  by  means 
of  equation  (124)  the  order  of  magnitude  of  ney  the  num- 
ber of  electrons  crossing  a  square  cm.  per  second  from  one 
side  to  the  other  in  a  metal.  Thus  for  example  in  the  case 
of  copper  we  have 


C  =  .7198  cal.   Sme  =  4.808X10-24  caL, 
and 

\  3  =  2.25XlO-8cm., 
which  gives 

ne  =  9.15X!030. 

It  is  of  interest  to  compare  this  number  with  the  number  of 
molecules  crossing  a  square  cm.  from  one  side  to  the  other 
per  second  in  a  liquid  or  dense  gas.  Thus  in  the  case  of 
CO2  at  40°  C.  under  a  pressure  of  200  atmos.  this  number  is 
equal  to  7.8X1026  according  to  Table  VIII.  The  density 
of  the  C02  under  these  conditions  is  about  equal  to  unity. 
The  value  of  lce  according  to  the  foregoing  considerations 
would  increase  with  increase  of  temperature  in  the  case 
of  the  pure  metals,  since  they  expand  with  increase  of  tem- 
perature, and  the  increase  may  probably  be  taken  approxi- 
mately equal  to  the  increase  in  the  distance  of  separation 
of  the  atoms.  Sme  is  very  likely  practically  independent  of 
the  temperature.  Therefore  for  the  cases  that  C  decreases 
with  increase  of  temperature  this  very  likely  also  holds  for 
ne  according  to  equation  (124).  Some  of  the  metals  falling 


168 


THE  NATURE  OF  MOLECULAR  MOTION 


into  this  category  are:    lead,   cadmium,   iron,   silver,   bis- 
muth, zinc,  and  tin. 

Other  investigations  having  a  bearing  on  the  electrons 
in  a  metal  will  be  found  in  Section  42. 

40.  The  Coefficient  of  Diffusion  of  a  Substance. 

If  the  constituents  of  a  mixture  of  substances  are  not 
evenly  distributed,  diffusion  from  one  part  to  another 
will  take  place  till  equilibrium  is  obtained.  The  primary 


Distance 
FIG.  16.    : 

cause  of  diffusion  is  the  motion  of  translation  of  the  mole- 
cules. This  will  at  once  appear  from  the  consideration  of 
a  mixture  whose  constituents  r  and  e  are  distributed  accord- 
ing to  the  curves  shown  in  Fig.  16,  the  concentration  being 
supposed  uniform  in  planes  parallel  to  the  concentration 
axes.  Molecules  are  projected  from  each  element  of  volume 
of  the  mixture.  The  portion  of  the  mixture  between  the 
planes  ab  and  cd  will  therefore  receive  more  molecules  r 
from  the  portion  of  the  mixture  between  the  planes  cd  and 
ef,  on  account  of  the  concentration  gradient  of  the  mole- 
cules r,  than  vice  versa.  Thus  on  the  whole  a  migration 
of  molecules  r  in  the  direction  of  decrease  of  concentration 
would  take  place.  Similarly  it  follows  that  the  molecules 


THE  COEFFICIENT  OF  DIFFUSION  169 

e  on  the  whole  migrate  in  the  direction  of  decrease  of  con- 
centration of  the  molecules  e,  or  in  the  opposite  direction 
to  the  molecules  r.  If  dr  denote  the  number  of  molecules  r 
which  on  the  whole  are  transported  across  a  square  cm., 

and  —  c  denotes  the  concentration  gradient,  we  have 
8x 

8r  =  Dr.,   ......     (125) 


where  DT  denotes  the  coefficient  of  diffusion  of  the  molecules 
r.  When  the  concentration  gradient  is  unity  Dr=  8r.  It 
is  evident  that  the  coefficient  of  diffusion  should  depend  on 
the  density  of  the  mixture,  the  ratio  of  the  masses  of  the 
constituents,  and  the  concentration  gradient,  at  the  point 
the  coefficient  is  measured.  It  is  found,  however,  to  be 
approximately  independent  of  the  concentration  gradient 
over  a  considerable  range  of  gradients. 

41.  The  Mean  Diffusion  Path  of  a  Molecule  in 
a  Mixture. 

The  path  of  a  molecule  in  a  mixture  in  migrating  from 
one  place  to  another  is  undulatory  in  character  on  account 
of  the  interaction  between  the  molecules  due  to  the  existence 
of  molecular  forces  and  molecular  volume.  It  differs  the 
more  from  a  number  of  straight  lines  of  various  lengths 
joined  consecutively  together  at  various  angles,  the  greater 
the  density  of  the  substance.  The  effect  of  this  path  on  the 
rapidity  of  the  diffusion  of  a  molecule  from  one  place  to 
another  may  be  represented  by  another  path  along  which 
the  molecule  is  supposed  to  move,  consisting  of  straight 
lines  joined  together  and  lying  near  the  path,  and  inci- 
dentally intersecting  it,  as  shown  by  Fig.  14.  In  supposing 
that  a  molecule  passes  along  its  representative  path  it  is 
obvious  of  course  that  in  general  the  molecule  at  any  in- 
stant does  not  occupy  its  actual  position.  But  this  does 


170  THE  NATURE  OF  MOLECULAR  MOTION 

not  matter  when  we  are  dealing  with  the  behavior  of  a  large 
number  of  molecules  as  a  whole.  The  straight  lines  consti- 
tuting the  representative  path  will  be  called  the  diffusion 
free  paths  of  the  molecule.  The  average  length  of  these 
paths  is  left  arbitrary  to  a  certain  extent  by  their  defini- 
tion, and  we  will  therefore  impose  the  conditions  that  the 
sum  of  the  representative  free  paths  between  any  two  points 
a  considerable  distance  apart  is  equal  in  length  to  the  cor- 
responding actual  path,  and  that  each  direction  of  a  free 
path  of  given  length  is  equally  probable.  These  conditions 
completely  determine  the  magnitude  of  the  mean  free  path, 
as  will  appear  from  the  next  Section.  These  paths  are  not 
equal  in  magnitude  but  are  grouped  about  the  mean  free 
path  ld  according  to  the  law  given  in  Section  31. 

42.  Formulae  Expressing  the  Coefficient  of  Dif- 
fusion in  Terms  of  Other  Quantities. 

Let  us  consider  a  heterogeneous  mixture  of  molecules 
r  and  e  in  which  the  molecules  are  uniformly  distributed 
in  planes  parallel  to  the  plane  AB  in  Fig.  15,  but  non-uni- 
formly  distributed  at  right  angles  to  the  plane  so  that  a 
diffusion  of  molecules  r  towards  the  parallel  plane  CD  takes 
place,  while  a  diffusion  of  molecules  e  takes  place  in  the 
opposite  direction.  We  may  suppose,  to  simplify  the  rea- 
soning, that  the  molecular  paths  which  cut  the  plane  EF 
in  migrating  towards  the  plane  CD,  begin  their  journey  in 
the  former  plane  at  the  same  point.  If  nr  denote  the  total 
number  of  molecules  r  crossing  the  plane  EF  per  square 
cm.  from  one  side  to  the  other,  which  is  equal  to  the  num- 
ber of  times  the  plane  is  cut  by  the  corresponding  free  paths, 
the  number  whose  paths  make  an  angle  8  with  a  perpendic- 
ular to  the  plane  is  similarly,  as  in  Section  34,  given  by 

nr  sin  6  -  d6. 


MOLECULAR  MIGRATION  ACROSS  A  PLANE         171 

The  number  of  the  foregoing  molecules  whose  paths  lie 
between  z  and  z-\-dz  is  equal  to 

Z     _£L 
nrsin  e-dd-^-e  I8r-dz, 

*    6r 

according  to  Section  31,  where  lsr  denotes  the  mean  dif- 
fusion path  of  the  molecules  r.  This  number  of  molecules, 
and  the  corresponding  number  of  molecules  moving  in  the 
opposite  direction,  which  we  may  suppose  begin  their  journey 
in  the  plane  GH  in  Fig.  15,  are  respectively  proportional 
to  the  molecular  concentrations  of  the  molecules  r  in  the 
planes  EF  and  GH.  If  Nr  denotes  the  concentration  in 
the  plane  EF,  the  concentration  in  the  plane  GH  is  evidently 

.  dNr      ,         dNr  •     ,, 

Nr—  z  cos  0-— — ,  where  — :—  is  the  concentration  gradient 
dx  dx 

measured  in  the  direction  of  increase  of  concentration. 
Therefore  the  loss  in  molecules  r  in  the  plane  EF  is  equal  to 
the  difference  in  the  foregoing  concentrations  divided  by 
NT,  and  multiplied  by  the  preceding  number  of  molecules, 
which  gives  for  the  loss 

1     dNr  Z2     ~Tfo 

TT? — r~rwr  sin  8  cos  B^—e      r  -dz-  dd. 
Nr  dx  Pi, 

The  total  loss  of  molecules  r  in  the  plane  is  obtained  by 
integrating  the  foregoing  expression  from  0  to  oo  with  respect 
to  the  free  path  z,  and  from  0  to  7r/2  with  respect  to  the  angle 
6.  On  referring  to  similar  integrals  in  Sections  34  and  38 
it  will  be  evident  that  the  integral  in  question  is  equal  to 

rirhr  dNr 
Nr    dx' 

Similarly  it  can  be  shown  that  the  gain  of  molecules  e  in  the 
plane  EF  is  equal  to 

Tvfife'1 


172  THE  NATURE  OF  MOLECULAR  MOTION 

where  the  concentration  gradient  of  the  molecules  e  is  meas- 
ured in  the  direction  of  their  increase  of  concentration, 
and  thus  in  the  opposite  direction  of  the  gradient  of  the 
molecules  r.  The  total  loss  of  molecules  r  and  e  in  the  plane 
EF  is  therefore  equal  to 


~Wr~~dx'     ~N7~dx' 

which  may  be  written  MT—Me.  When  the  molecular  paths 
are  in  their  original  positions  the  foregoing  number  repre- 
sents, on  the  whole,  the  gain  in  molecules  on  the  lower 
side  of  the  plane  EF.  But  the  space  occupied  by  these 
molecules  is  not  zero.  A  portion  of  the  mixture  must  there- 
fore be  transported  bodily  across  the  plane  in  the  oppo- 
site direction  to  make  room  for  the  foregoing  molecules. 
Let  Pv  denote  the  volume  of  this  portion  of  the  mixture. 
The  total  number  of  molecules  r  transported  on  the  whole 
across  the  plane  per  square  cm.  in  the  direction  of  decrease 
of  concentrtaion  gradient,  which  is  equal  to  the  diffusion 
6r,  is  accordingly  given  by 

Hrhr  dNr          P,Nr 


Nr      dx        Ne+Nr 

Let  #r  and  &e  denote  the  external  molecular  volumes  of  a 
molecule  r  and  of  a  molecule  e  respectively  in  the  mixture, 
i.e.,  the  decrease  in  the  volume  of  the  mixture  when  a  mole- 
cule r  or  e  at  constant  pressure  is  removed.  These  quanti- 
ties are  evidently  connected  by  the  equation 


The  quantity  Pv  is  therefore  given  by  the  equation 


THE  GENERAL  DIFFUSION  EQUATION  173 

or 

'+ 


On  substituting  this  expression  for  Pv  in  the  preceding  dif- 
fusion equation,  and  then  substituting  the  expressions  for 
Me  and  M  r,  the  equation  becomes 


.  _  e  rhrNe  dN  T    nel&eNT  dN  e 

~Nr&r  +  Ne#t\      Nr        dx  :""*        Ne         dx 

which  is  a  fundamental  form  of  the  diffusion  equation. 
The  rate  of  diffusion  of  the  molecules  e,  which  takes  place 
in  the  opposite  direction  to  the  molecules  r,  is  obtained  on 
interchanging  the  suffixes  r  and  e  in  the  foregoing  equation. 
It  will  readily  be  seen  that  the  rates  of  diffusion  for  the 
two  different  kinds  of  molecules  may  be  written  5r  =  K&e,  and 
de  =  K&r,  and  hence  the  ratio  of  the  rates  of  diffusion  is 
given  by 

H  ........  (127) 

In  the  case  of  a  liquid  mixture  the  quantities  &,  and  #c  are 
not  equal  to  each  other.  They  may  easily  be  measured  in 
practice  since 

8v  dv 


where  v  denotes  the  volume  of  the  mixture. 

In  the  case  of  a  gaseous  mixture  de  =  $r,  and  Ne+Nr=* 
Cer  a  constant  since  the  pressure  is  everywhere  the  same. 
Equation  (126)  in  that  case  becomes 


rsre   ,       erT  n  OQ>. 

-~    ~~ 


If  the  concentration  of  one  of  the  constituents  of  the 
mixture  is  small  in  comparison  with  that  of  the  other,  say 


174  THE  NATURE  OF  MOLECULAR  MOTION 

of  the  molecules  r  in  comparison  with  the  molecules  e, 
equation  (126)  becomes 


The  rate  of  diffusion  is  then  independent  of  $r  and  $e> 

The  forms  of  the  foregoing  equations  may  be  modified 
by  considering  similarly  as  in  the  four  previous  Sections 
a  representative  mixture  which  has  the  same  rate  of  diffusion, 
expansion  pressure,  and  molecular  volumes  $r  and  $e,  but 
whose  molecules  do  not  possess  an  apparent  molecular 
volume  6  whose  defining  property  according  to  Section  19 
is  that  a  change  in  its  value  produces  a  change  in  the  external 
pressure  without  changing  the  total  average  velocity.  It 
should  be  carefully  noted  that  the  quantities  b  and  d  do  not 
mean  the  same  thing,  one  may  be  zero  when  the  other  is 
not.  The  former  quantity  represents  the  obstruction  the 
molecules  present  to  each  other's  motion,  while  the  latter 
quantity  represents  the  change  in  the  external  volume  of 
a  mixture  on  adding  a  molecule.  It  will  be  convenient 
now  to  replace  the  volume  v  by  the  concentration  Ner 
similarly  as  in  Sections  35  and  39.  The  molecular  free  paths 
of  the  molecules  r  and  e  may  then  be  expressed  similarly 
as  before  by  the  equations 


vr      NK' 


and 


Vr  — 


\ 
) 


.    .     .     .     (130) 


where  vr  denotes  the  volume  of  the  mixture  containing  a 
gram  molecule  of  molecules  r,  Ner  the  total  concentration 
of  the  molecules  r  and  e,  b'r  the  apparent  molecular  volume 
of  the  mixture  which  appears  to  obstruct  the  motion  of  the 
molecules  r,  K.'  &  the  characteristic  factor  of  the  diffusion 


THE  DIFFUSION  EQUATION  FOR  A  GAS  175 

free  path  of  a  molecule  r  when  the  mixture  is  in  the  gaseous 
state,  and  <£'  5r  the  diffusion  interference  function  with  respect 
to  a  migrating  molecule  r,  while  the  other  symbols  have 
similar  meanings.  As  a  first  approximation  we  may  write 
similarly  as  before 


where  N  denotes  the  number  of  molecules  in  a  gram  mole- 
cule of  a  pure  substance,  and  </>'  8r  and  </>'  Se  are  functions  of 
T  and  Ner,  but  which  are  insensitive  to  variations  in  Ner. 
On  applying  equations  (130)  to  the  gaseous  state  they 
become 

I    =K'   — 

and 

N 


Substituting  for  ldr  and  I8e  from  these  equations  in  equation 
(128)  and  obtaining  expressions  for  nr  and  ne  by  means  of 
equation  (21)  the  equation  becomes 


N         RT{NeK'8  r       NrK'8e\dNr 

°r~  *f2~\lo~\  —  7=~i  --  T^r^T"'      •      • 
N2er\   3  \\/mr     Vme\  dx 

since  Ne-}~Nr  =  Ner,  vrmarNr  =  mr,  and  vemaeNe  =  me,  where 
mar  and  mae  denote  the  absolute  and  mr  and  me  the  relative 
molecular  'weights  of  the  molecules  r  and  e  respectively. 
This  equation  applies  to  the  gaseous  state  and  gives  the 
relation  between  the  characteristic  quantities  Krdr  and 
K'  Se  which  are  functions  of  the  temperature  and  the  nature 
of  the  molecules. 


176  THE  NATURE  OF  MOLECULAR  MOTION 

Since  lSr=K'SrN/N€r  when  the  mixture  is  in  the  gaseous 
state,  it  will  be  evident  from  an  inspection  of  Fig.  14,  which 
shows  the  relation  between  the  free  paths  of  a  molecule 
and  its  actual  path,  that  l/lSn  or  !/*'&.  is  a  measure  of  the 
chance  of  the  resultant  force  on  a  migrating  molecule  passing 
through  a  maximum  or  minimum  and  changing  its  direction. 

When  the  mixture  is  in  the  gaseous  state  and  the  con- 
centration of  one  set  of  molecules  is  relatively  small,  another 
interesting  significance  may  be  attached  to  the  quantity 
K'  8r.  Equation  (129)  then  applies,  and  it  indicates  that  for 

the  same  concentration  gradient  per  molecule,  or   „     f  = 

constant,  the  foregoing  quantity  is  proportional  to  dr/nr. 
Therefore,  since  nr  denotes  the  number  of  molecules  crossing 
per  second  per  square  cm.  a  plane  situated  at  right  angles 
to  the  direction  of  diffusion,  and  dr  denotes  the  rate  the 
molecules  diffuse  across  the  plane,  the  quantity  K' Sr  is  a 
measure  of  the  chance  of  a  molecule  which  has  crossed  the 
plane  from  one  side  to  the  other,  to  remain  on  the  latter 
side. 

The  quantities  &  Sr  and  $'  Se  in  the  foregoing  equations 
express  the  effect  of  the  molecular  interference  of  the  mixture 
on  two  interacting  molecules  in  changing  the  value  of  5r, 
which  is  not  expressed  by  the  other  quantities  which,  are 
similarly  affected. 

The  values  of  nr,  ne,  b'r,  and  b'e  in  the  foregoing  equations 
may  be  found  by  the  methods  of  Section  29,  or  by  those 
given  in  Section  46.  The  values  thus  obtained  may  be  cor- 
rected according  to  Maxwell's  distribution  of  molecular 
velocities  in  the  way  described.  The  values  of  the  other 
quantities  which  cannot  be  measured  directly  may  be 
determined  by  methods  which  will  now  be  described. 

In  the  case  of  a  dilute  solution  of  molecules  r  in  e  the 
value  of  ldr  is  immediately  given  by  equation  (129)  since 
8r  may  be  measured  directly.  If  the  coefficient  of  diffusion 


CHARACTERISTIC  FUNCTION  APPROXIMATION     177 

for  the  same  kind  of  mixture  in  the  gaseous  state  were 
measured,  or  otherwise  were  known,  the  value  of  K' ST  would 
be  given  by  equation  (133)  to  which  equation  (129)  can 
be  reduced.  The  value  of  <£'  Sr  could  then  be  obtained  from 
equations  (130). 

In  the  case  of  a  mixture  in  which  one  of  the  constituents 
is  not  small  in  comparison  with  the  other  the  quantities 
K!  Sr  and  K'  8e  may  be  determined  by  means  of  equation  (131). 
Thus  we  may  write 

1  ai.l 

K'ST      Ner 

and 


similarly  as  in  Section  35,  and  substitute  from  these  equa- 
tions for  the  foregoing  quantities  in  equation  (131).  The 
three  constants  ar,  ae,  and  are,  at  constant  temperature 
may  be  determined  by  applying  the  resultant  equation 
to  three  diffusing  gaseous  mixtures  of  different  relative 
concentrations,  which  furnishes  three  simultaneous  equa- 
tions. The  values  of  K'  Sr  and  K'  de  may  then  be  calculated 
for  any  relative  concentration  of  the  molecules  by  means 
of  the  foregoing  two  equations. 

An  interesting  special  case  of  such  calculations  corresponds 
to  Ne—Q.  The  resultant  value  of  K'  Sr  may  be  used  to  calcu- 
late the  coefficient  of  diffusion  of  a  molecule  r  in  a  gas  of 
the  same  kind  (a  quantity  which  cannot  be  measured 
directly)  by  means  of  equation  (131)  putting  JVe=0. 

The  values  of  K.'  Sr  and  K'  Se  obtained  by  the  foregoing 
method  would  not  be  corrected  for  the  distribution  of 
molecular  velocities.  If  this  correction  is  carried  out  accord- 
ing to  Maxwell's  law  each  value  has  to  be  multiplied  by 
1.085. 


178  THE  NATURE  OF  MOLECULAR  MOTION 

The  quantities  <j>' 5r  and  </>'  8e  may  now  be  determined  by 
means  of  equation  (126)  on  substituting  for  <t>'5r,  <J>'ar, 
I5n  and  I8e  in  the  equation  from  the  equations  defining  these 
quantities,  and  for  0'  Sr  and  cf>f  Se  from  the  equations 

/V 

and 


deduced  similarly  as  similar  equations  in  Section  35.  An 
equation  is  obtained  containing  the  four  constants  be,  br, 
bree,  and  brre  at  constant  temperature.  Therefore  on  apply- 
ing the  equation  to  four  dense  diffusing  mixtures  of  dif- 
ferent relative  concentrations  of  the  molecules  r  and  e  four 
simultaneous  equations  will  be  obtained  from  which  these 
constants  can  be  determined.  The  values  of  0'5r  and  </>'  8e 
may  then  be  calculated  for  any  relative  concentration  of 
the  molecules  r  and  e  in  the  mixture  by  means  of  the  fore- 
going equations.  An  interesting  special  case  of  such  a 
calculation  is  that  corresponding  to  Ne  =  0. 

The  number  of  times  Ns  a  molecule  passes  over  its 
mean  free  diffusion  path  ls,  which  is  inversely  proportional 
to  the  number  of  times  the  force  on  a  molecule  passes 
through  a  maximum  and  changes  its  direction  of  motion, 
is  given  by 

tf,-f',      ......     (132) 

^5 

where  Vt  denotes  the  total  average  velocity  of  a  molecule. 
This  equation  follows  from  the  fact  that  the  representative 
path  of  a  molecule  is  equal  in  length  to  the  actual  path. 

(a)  As  an  application  of  the  foregoing  investigation  let 
us  consider  the  diffusion  of  molecules  r  and  e  in  the  gaseous 
state  into  each  other.  If  the  concentration  of  the  mole- 


DIFFUSION  IN  DILUTE  MIXTURE  OF  GASES       179 

cules  r  is  small  in  comparison  with  that  of  the  molecules 
e  the  coefficient  of  diffusion  Dr  of  the  molecules  r  is  accord- 
ing to  equations  (125),  (129),  and  (130),  given  by 


Now  according  to  equations  (35)  and  (8) 

~3         3"\"^7' 
and  the  foregoing  equation  may  therefore  be  written 


(134) 


The  coefficient  of  diffusion  is  therefore  inversely  propor- 
tional to  the  total  molecular  concentration,  since  K.'  Sr  is 
independent  of  the  volume  of  the  mixture.  This  result  is 
borne  out  by  experiment. 

The  foregoing  equation  may  also  be  written 


.     .     .     (135) 

mr 

since  ]Ver  =  7.46Xl015p/Tr  according  to  equation  (15), 
N  =  6.2X1023,  and  R  =  8.315  X107,  where  p  denotes  the 
external  pressure.  This  equation  was  used  to  calculate 
the  values  of  KS  given  in  Table  XX  for  the  inter-diffusion 
of  a  number  of  different  gases,  where  for  convenience  and 
simplicity  KS  is  now  written  for  Kfdn  the  values  of  D  used  cor- 
responding to  p  =  760  cm.  of  mercury.  The  directions  of 
the  diffusion  to  which  the  coefficients  in  the  Table  refer 
are  indicated  by  arrow  heads. 

It  will  be  seen  on  inspecting  the  table  that  KS  increases 
with  increase  of  temperature  for  the  same  two  diffusing 
gases.  The  chance  of  the  resultant  force  on  a  migrating 


180 


THE  NATURE  OF  MOLECULAR  MOTION 


molecule  per  unit  length  of  path  passing  through  a  maximum 
and  changing  in  direction  is  thus  decreased  by  an  increase 
of  temperature.  Actually  this  means  that  some  of  the 
smaller  maxima  or  bends  of  the  molecular  path  are  more  or 
less  smoothed  out  by  an  increase  of  temperature.  This 
would  happen  because  the  time  any  pair  of  interacting  mole- 
cules are  under  each  other's  influence  is  decreased  by  an 
increase  of  temperature,  since  this  increases  the  molecular 
velocities,  and  hence  the  amount  of  deflection  each  mole- 
cule undergoes  is  decreased. 


TABLE   XX 


t°  C. 

D 

*510'° 

D 

Kg   1010 

H20-+C02 

H20->H2 

0 
92.4 

.132 
.2384 

2.84 
3.31 

.687 
1.179 

14.8 
16.3 

CH4O-»CO2 

CH4O—  »H2 

0 

49.6 

.0880 
.1234 

2.52 
2.75 

.5001 
.6738 

14.3 
15.0 

C6H6^C02 

C6H6-»H2 

0 
45 

.0527 
.0715 

2.36 
2.54 

.294 
.3993 

13.1 
14.2 

C9H18O2-»CO2 

C9H1802-»H2 

0 

97.8 

.0305 
.0568 

1.94 
2.29 

.1724 
.3177 

11.0 
12.8 

PROPERTIES  OF  THE  CHARACTERISTIC  FUNCTION    181 

It  appears  also  that  for  the  same  medium  the  value  of 
KS  decreases  with  an  increase  of  molecular  weight  of  the 
diffusing  molecule.  This  indicates  that  the  chance  of  the 
resultant  force  on  a  migrating  molecule  per  unit  length 
of  its  path  passing  through  a  maximum  and  changing  in 
direction  is  increased  by  an  increase  in  the  molecular  weight 
of  the  molecule.  The  relative  change  in  KS,  it  will  be 
noticed,  is  considerably  smaller  than  the  relative  change  in 
the  molecular  weight.  Thus  an  increase  in  molecular 
weight  acts  in  the  opposite  direction  to  an  increase  in  tem- 
perature on  the  value  of  K,,,  as  we  might  expect.  It  is 
evident  that  an  increase  in  molecular  weight  increases  the 
time  the  interacting  molecules  are  under  each  other's  influ- 
ence, since  this  is  attended  by  a  decrease  in  their  velocities, 
and  the  force  that  they  exert  upon  each  other  is  also  in- 
creased. The  values  in  the  Table  are  not  corrected  for 
Maxwell's  law  of  distribution  of  molecular  velocities  the 
correction  corresponding  to  the  introduction  of  the  factor 
1.085. 

If  the  quantities  ne  and  nr  in  equation  (128),  which  applies 
to  the  gaseous  state,  are  expressed  in  terms  of  the  correspond- 
ing molecular  velocities,  the  equation  assumes  the  form 
usually  given  in  treatises  on  the  Kinetic  Theory  of  Gases. 
The  quantities  lsr  and  lse  are  supposed  to  refer,  however, 
to  the  free  paths  of  molecular  collision  of  the  molecules  r 
and  e  respectively,  and  therefore  have  not  the  same  funda- 
mental meaning  given  to  them  in  this  book. 

(6)  An  interesting  application  of  equation  (129)  may 
be  made  in  connection  with  the  electrons  in  a  metal.  We 
may  write  Ne  =  Kene,  where  Ne  denotes  the  number  of 
free  electrons  per  cubic  cm.  in  a  metal,  and  ne  the  number 
crossing  a  square  cm.  from  one  side  to  the  other  in  all  direc- 
tions per  second.  A  change  in  Ne  evidently  produces  a 
change  in  ne  through  the  increase  in  concentration  of  the 
electrons  and  the  change  in  their  interaction  upon  each  other. 


182  THE  NATURE  OF  MOLECULAR   MOTION 

It  will  not  be  difficult  to  see  that  the  change  in  ne  from  the 
former  .cause  is  large  in  comparison  with  that  due  to  the 
latter  when  the  change  in  concentration  is  small.  We  may 
therefore  consider  Ke  a  constant  for  small  changes  of  Ne. 
Equation  (129)  applied  to  the  diffusion  of  electrons  in  a 
metal  may  therefore  be  written 

.         dne 


Since  —  ne  is  equal  to  the  partial  expansion  pressure  Pe  of 

the  electrons  according  to  Section  20  the  foregoing  eauation 
may  be  written 


where  —7-^  is  evidently  the  force  acting  on  a  cubic  cm.  of 

diffusing  electrons.  We  may  therefore  suppose  that  the 
electrons  are  uniformly  distributed  and  this  force  produced 
by  an  electric  field  X,  so  that 

dPe 

-r-, 

ax 

where  e  denotes  the  electric  charge  on  an  electron,  and  the 
equation  therefore  given  the  form 

2l8eXeNe 

8e=     -I—  . 

The  current  /  produced  by  the  electric  field  is  given  by 


where  k  denotes  the  electric  conductivity  of  the  material; 
and  the  foregoing  two  equations  therefore  give 


2. 


THE  CONCENTRATION  OF  ELECTRONS  IN  METALS    183 

where  me  denotes  the  mass  of  an  electron  relative  to  the 
hydrogen  atom,  and  A  =  5.Q87XlQ~20V~Tme.  This  equa- 
tion may  be  used  to  calculate  the  number  of  electrons  per 
cubic  cm.  of  a  metal. 

The  value  of  the  free  diffusion  path  lSe  of  an  electron  in 
a  metal  is  probably  in  the  main  governed  by  the  presence 
of  the  atoms,  and  as  a  first  approximation  may  therefore 
be  taken  equal  to  the  distance  of  separation  of  the  atoms. 
Approximate  values  of  Ne  may  thus  be  obtained.  Thus 
for  example  in  the  case  of  copper  at  0°  C.  &  =  7.7X10~4 
E.M.U.,  and  since  m«=.001,  e=1.6X10~20  E.M.U.,  T  = 
273°,  and  lde  =  2.25XlO~8  approximately,  we  obtain 

Ne  =  2  X  1024  approximately. 

Equation  (136)  gives  definite  information  about  the 
variation  of  Ne  with  the  temperature.  Since  k  for  the  pure 
metals  varies  approximately  inversely  as  the  absolute 
temperature,  it  follows  from  the  equation  that 


Therefore,  since  lSe  can  only  increase  with  increase  of  tem- 
perature, the  value  of  Ne  decreases  with  increase  of  tem- 
perature in  the  case  of  the  pure  metals.  This  result  may 
of  course  not  hold  in  the  case  of  alloys  whose  conductivity 
usually  increases  with  increase  of  temperature. 

Experiment  shows  that  the  conductivity  of  a  metal 
is  greatly  affected  by  small  amounts  of  impurities.  These 
should  not  alter  the  value  of  lse  to  an  appreciable  extent. 
The  change  in  conductivity  must  therefore  be  due  according 
to  equation  (136)  to  a  change  in  Ne  induced  by  the  impuri- 
ties. This  is  probably  brought  about  by  the  impurities 
affecting  the  rate  of  dissociation  of  the  atoms  of  the  metal, 
and  therefore  the  corresponding  constant  of  mass-action. 


184 


THE  NATURE  OF  MOLECULAR  MOTION 


From  equations  (136)  and  (124)  we  obtain 

C     2.543  X  W-20nelceSmeVTme 
k  ~  Nel8ee2 

This  equation  gives  the  value  of  the  ratio 


(137) 


* 


the  quantities  C  and  k  can  be  measured  directly,  and  the 
remaining  quantities  are  known  constants.  The  ratio  C/k 
has  approximately  the  same  value  for  all  pure  metals  at 
the  same  temperature,  and  is  approximately  proportional 
to  the  absolute  temperature  T.  This  is  shown  by  Table 
XXI,  which  was  taken  from  Richardson's  Electron  Theory 

TABLE  XXI 


Material. 

Values  of 
C/k  at  18°  C. 

Temp.  Coef. 
of  Ratio. 

Copper  (pure) 

6.65  X1010 

3.9X10"3 

Silver  (pure) 

6.86  X1010 

3.7X10"3 

Gold  (pure)  
Nickel  (pure)    ...          

7.27  X1010 
6.99  X1010 

3.6X1Q-3 
3.9X10"3 

Zinc  (pure) 

7.05  X1010 

38X10"3 

Cadmium  (pure) 

7.06  X1010 

3  7X10~3 

Lead  (pure)  

7.15X1010 

4.0X10"3 

Tin  (pure)                        .    ... 

7.35  X1010 

3.4X10"3 

Aluminium 

636X1010 

43X10"3 

Platinum  (pure)  

7.53X10lft 

4.6X10"3 

Palladium   

7.54  X1010 

4.6X10"3 

Iron 

8.02  X1010 

4.3X10"3 

Bismuth 

9.46  X1010 

1.5X10"3 

of  Matter.  Since  /«/£««  is  very  probably  independent  of 
the  temperature,  and  this  is  also  likely  to  hold  approxi- 
mately for  Sme,  the  partial  specific  heat  of  the  electrons,  it 
follows  that  approximately 


APPROXIMATE  CONDUCTIVITY  EXPRESSION       185 

Another  expression  for  the  electric  conductivity  of  a 
metal,  which  is  well  known,  may  be  obtained  as  follows: 
Consider  a  metal  in  which  the  electrons  are  under  the 
action  of  an  electric  field  of  intensity  X.  Let  I  denote  the 
length  of  the  actual  path  of  an  electron  at  the  end  of  which 
it  possesses  the  same  amount  of  kinetic  energy  as  at  the 
beginning,  being  the  path  over  which  all  the  energy  imparted 
to  the  electron  by  the  electric  field  is  imparted  to  the  sur- 
rounding electrons  and  molecules  of  the  metal.  It  will 
be  noticed  that  again  we  define  molecular  path  without  the 
introduction  of  molecular  collision,  as  this  is  more  satisfac- 
tory. If  Vt  as  usual  denotes  the  total  average  velocity 
(Section  17)  of  an  electron,  the  time  t  it  takes  to  traverse 
the  distance  I  is  given  by  t  =  l/Vt.  The  effect  of  the  electric 
field  is  to  give  an  acceleration  equal  to  Xe/mae  to  the  electron, 
whose  absolute  mass  is  mae.  If  we  suppose  that  this  is 
unimpeded  along  the  path  of  the  electron,  it  will  have 

Xet 
a  component  velocity  equal  to  -  -  at  the  end  of  the  path. 

mae 

This  causes  a  drift  of  the  electrons  which  is  approximately 

Xet2 
equal  to  —  -  during  the  time  t.    The  average  velocity  of 


the  drift  is  therefore  "-  —  ,  or  -  —  —  .    The  electric  current 
2mae         2maeVt 

in  the  metal  is  therefore  given  by 


and  accordingly  the  electric  conductivity  k  by 


NJl  _N?<?1 

"'"oZ; Tr~aZ^.^~>          ....      V100^ 


•by  the  help  of  equation  (35). 


...  186  THE  NATURE  OF  MOLECULAR  MOTION 

If  the  supposition  is  made  that  I  =  lc,  and  that  the  electrons 
behave  as  if  they  were  in  the  perfectly  gaseous  state,*  the 
ratio  C/k  formed  from  equations  (138)  and  (105)  is  pro- 
portional to  the  absolute  temperature  T,  and  its  value 
agrees  approximately  with  that  found  by  experiment.  It 
is  hardly  likely,  however,  that  the  suppositions  made  are 
true,  since  the  theoretical  ratio  shows  great  deviations 
from  the  experimental  when  the  metals  contain  small 
amounts  of  impurities,  which  should  not  affect  the  supposi- 
tions made.  Moreover,  it  is  obvious  that  in  general  I  can- 
not be  equal  to  lc.  What  we  can  say  with  certainty  only 
is  that  the  theoretical  ratio  of  C/k  has  a  factor  of  T  which 
for  pure  metals  has  a  value  corresponding  to  the  electrons 
behaving  as  if  they  were  in  the  perfectly  gaseous  state, 
according  to  the  results  of  experiment. 

From  equations  (138)  and  (136)  on  eliminating  k  we 
have 

I  2.545X10-20\/rm^ 

l~ 


This  equation  identically  vanishes  if  we  assume  that  I  =  2l5e, 
and  that  Vt  is  given  by  equation  (8),  which  corresponds 
to  the  electrons  behaving  as  if  they  were  in  the  perfectly 
gaseous  state.  Thus  the  latter  assumption,  and  the  assump- 
tion that  l  =  lce  =  2lse,  satisfy  equations  (124),  (136),  and 
(138). 

Accordingly  the  assumption  that  lce  =  2lSe)  and  that  the 
electrons  bthave  as  if  they  were  in  the  gaseous  state,  give 
a  value  for  the  theoretical  ratio  of  C/k  expressed  by  equation 
(137)  which  agrees  with  the  facts.  But  the  assumptions  are 
not  likely  to  hold  for  the  same  reasons  as  stated  previously. 

*  Which  corresponds  to  n  being  given  by  equation  (21),  Sme  being 
equal  to  4.808  XlO~24  cal.,  and  taking  into  account  that  vNemae  =  me 
=  .001. 


INVERSE  FIFTH  POWER  LAW  OF  ATTRACTION     187 

The  expression  for  the  ratio  C/k  obtained  from  equa- 
tions (124)  and  (138)  applied  to  the  gaseous  state  is  the 
one  usually  given  in  connection  with  electric  conductivity, 
but  the  meanings  usually  attached  to  I  and  lce  are  not 
exactly  the  same  as  those  given  in  this  book.  The  ex- 
pression for  the  ratio  given  by  equation  (137)  is  mathe- 
matically more  fundamental,  as  will  easily  be  recognized. 

The  diffusion  of  gases  may  also  be  treated  from  another 
aspect,  which  involves  the  attraction  of  the  molecules  upon 
each  other. 

43.  Maxwell's  Expression  for  the  Coefficient  of 
Diffusion  of  Gases. 

If  a  molecule  may  be  regarded  simply  as  a  center  of 
forces  of  attraction  and  repulsion  the  coefficient  of  diffusion 
of  a  gas  into  another  gas  is  a  function  of  the  law  of  force 
between  the  molecules.  The  mathematics  involved  in 
finding  the  function  for  any  given  law  of  force  is,  however, 
of  a  very  complicated  character,  and  does  not  yield  expres- 
sions of  any  practical  use  except  in  one  case.  Maxwell* 
has  shown  that  if  the  attraction  between  two  molecules 
separated  by  a  distance  x  may  be  represented  by  the  expres- 
sion B/x5,  where  B  denotes  a  constant  depending  on  the 
nature  of  the  molecules,  the  coefficient  of  diffusion  DT  of 
a  gas  r  into  a  gas  e  is  given  by 


Pe 


)r      A     f  B 

Pc^lc 


(139) 


where  pr,  pr,   pe,  and  pe  denote  the  partial  densities  and 
pressures  respectively  of  the  molecules  r  and  e  whose  molec- 
ular weights  are  mf  and  me,  P  =  pr+Pe}  and  Ac  denotes  a 
*  "  Dynamical  Theory  of  Gases  "  Collected  Papers,  Vol.  II,  p.  36. 


.188    THE  NATURE  OF  MOLECULAR  MOTION 

numerical  constant.  The  attraction  between  two  molecules 
may  probably  over  a  certain  region  be  approximately 
represented  by  a  single  term  of  the  above  form  according 
to  Section  26.  Actually  the  force  between  two  molecules 
according  to  Section  14  is  expressed  by  a  number  of  terms 
which  is  probably  greater  than  three.  The  equation  has 
been  applied  to  gases  by  the  writer  *  to  calculate  their 
relative  coefficients  of  diffusion,  in  connection  with  investi- 
gations of  the  nature  of  the  forces  of  molecular  interaction. 
The  discussion  of  the  results  is  therefore  reserved  for 
another  place. 

In  the  previous  Sections  the  diffusion,  viscosity,  conduc- 
tion of  heat,  was  investigated  without  any  reference  to  the 
exact  nature  of  the  law  of  force  of  molecular  interaction. 
The  free  paths  introduced  have  not  the  direct  physical 
significance  associated  with  the  paths  defined  by  the  old 
method  of  molecular  collision.  But  we  have  seen  that 
some  other  important  physical  significance  can  be  attached 
to  each  path,  and  to  the  more  important  component  factor 
K  about  which  really  the  interest  of  each  kind  of  path  cen- 
ters. This  procedure,  there  can  be  no  doubt,  is  more  funda- 
mental than  that  involving  molecular  collision,  and  more- 
over it  is  mathematically  quite  sound,  besides  being  very 
much  simpler.  The  constants  involved  in  the  various 
expressions  obtained  in  this  way  are  obviously  functions  of 
the  law  of  molecular  attraction  and  repulsion.  The  results 
obtained  will  be  used  in  the  next  chapter  to  interpret 
various  aspects  of  Brownian  motion,  and  the  diffusion  and 
mobility  of  particles.  The  molecular  free  paths  will  also 
be  given  extended  forms  along  the  lines  worked  out  in  the 
previous  Sections,  by  the  aid  of  which  useful  and  interesting 
information  may  be  obtained  about  molecular  motion  under 
various  conditions, 

*Phil  Mag.,  May,  pp.  783-809,  1910. 


CHAPTER    IV 

MISCELLANEOUS  APPLICATIONS,  CONNECTIONS,  AND 
EXTENSIONS  OF  THE  RESULTS  OF  THE  PREVIOUS 
CHAPTERS 

44-  The  Direct  Observation  of  some  of  the  Quan- 
tities depending  on  the  Nature  of  the  Motion  of  a 
Molecule  in  a  Substance,  and  their  Use. 

In  Section  4  we  have  seen  that  the  motion  of  transla- 
tion of  a  colloidal  particle  in  a  liquid  may  be  of  such  a 
magnitude,  depending  on  the  size  of  the  particle,  that  it 
can  conveniently  be  observed  directly  by  means  of  the 
ultra  microscope.  The  motion  is  oscillatory  in  nature, 
and  takes  place  in  haphazard  directions,  giving  rise  to  a 
migration  of  the  particle  which  is  also  oscillatory  and  hap- 
hazard in  character.  If  the  solution  is  given  a  motion  at 
right  angles  to  the  microscope  by  passing  a  stream  of  the 
solution  through  the  containing  vessel,  the  motion  of  each 
particle  as  it  appears  to  the  eye  is  the  resultant  of  the  motion 
of  the  particle  and  that  of  the  liquid,  and  thus  the  particle 
appears  to  traverse  an  undulatory  or  wavy  curve.  This 
method  of  observation  was  introduced  by  Svedberg*, 
who  accordingly  speaks  of  the  average  amplitude  A\  and 
wave-length  X  of  the  apparent  path  of  the  particle.  Fig. 
17  shows  as  an  illustration  some  curves  that  he  obtained 
in  this  way.  The  left-hand  side  of  the  figure  shows  cases 

*Zs.  f.  Eledroch.,  12,  1906,  pp.  853-909;  Zs.  f.  Phys,  Chem.,  71, 
1910;  p.  571. 

189 


190    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 


of  motion  of  particles  as  they  would  appear  to  the  eye  if 
the  fluid  were  at  rest,  while  the  right-hand  side  shows  the 
curves  they  would  trace  out  in  the  plane  of  observation 
on  giving  a  parallel  motion  to  the  fluid.  If  the  actual  motion 
of  a  particle  in  a  liquid  at  rest  were  along  a  number  of  straight 
lines  joined  together  consecutively  at  different  angles  into 
a  continuous  line,  and  this  motion  were  compounded  with 
a  uniform  motion  in  a  given  direction,  the  motion  of  the 


particle  projected  on  to  a  plane  would  appear  as  a  curve  of 
a-  similar  character,  namely  consisting  of  straight  lines 
joined  together  at  various  angles.  This  is  not  observed 
in  practice,  however,  and  it  follows  therefore  that  the  motion 
of  a  particle  is  not  suddenly  changed  in  direction  at  cer- 
tain points  only  along  <  its  path,  but  continually  so  more 
or  less,  showing  that  it  is  continually  under  the  influence 
of  the  surrounding  molecules. 


THE  PROJECTION  OF  A  PATH  ON  TO  AN  AXIS      191 

If  a  point  is  chosen  in  the  vicinity  of  each  bend  of  the 
actual  path  of  the  particle,  and  each  consecutive  pair  of 
points  joined  by  a  straight  line  subject  to  the  conditions 
that  the  sum  of  the  lengths  of  the  lines  between  two  points 
a  considerable  distance  apart  is  equal  in  length  to  the  cor- 
responding path,  and  each  direction  of  a  line  in  space  is 
equally  probable,  each  line  corresponds  to  a  diffusion  free 
path  of  the  particle  according  to  Section  41. 

The  projection  of  a  diffusion  path  on  to  a  plane  can 
be  shown  to  reduce  it  on  the  average  in  the  ratio  7r/4,  since 
it  may  point  in  any  direction.  Thus  if  ls  denotes  the  length 
of  the  mean  diffusion  path  of  the  particles  making  an  angle 
0  with  a  line  at  right  angles  to  the  plane  of  observation, 
and  no  particles  pass  through  each  point  per  second,  the 
length  of  the  projection  of  these  paths  on  to  the  plane  is 
equal  to 

/""T  /*V 

I    2?r/gsin  6-ls'dd-  .  °19'l$smd  =  ^       sin2  0-dd 
Jo  47iV  2  Jo 

'S^O  I       •         n  l     TT^Oy  '5^0     I  •      O    n       ls\ 

=  -jr-  sin  0  •  cos  0     +  —=r  Is  — s-       sin2  0  •  dB 
*    L  Jo       ^          I  Jo 

"T^;.:2jl  Sin2^' 

and  thus  is  equal  to  ^pZ*,  or  J5  for  a  single  path.     The 

average  projection  p  of  a  diffusion  path  on  to  an  axis  in  the 
plane  of  observation  can  now  be  shown  to  be  given  by 

4 


or 


192    MISCELLANEOUS  APPLICATIONS    CONNECTIONS 

where  <t>  denotes  the  angle  the  projection  -ls  makes  with 

a  line  at  right  angles  to  the  foregoing  axis. 

The  projection  p  is  evidently  somewhat  greater  than 
twice  the  average  amplitude  A\  of  the  curve  traced  out  in 
the  plane  of  observation  by  the  projected  motion  of  a  particle 
when  the  solution  is  given  a  motion  at  right  angles  to  the 
axis  of  projection  of  p.  It  is  evident  that  the  closer  the 
actual  path  of  the  particle  resembles  a  number  of  straight 
lines  joined  together  at  various  angles  the  smaller  is  this 
difference.  Its  magnitude  is  impossible  to  determine  theo- 
retically, but  it  is  probably  safe  to  say  that  it  is  not  likely 
to  be  greater  than  10  per  cent,  probably  it  is  often  much  less. 
Thus  as  a  first  approximation  we  may  take  ls  equal  to  4A  \. 

It  is  possible,  however,  to  determine  the  exact  average 
value  of  the  mean  diffusion  path  ld  from  curves  of  the 
nature  shown  in  Fig.  17.  The  average  wave  length  X  of 
*the  curve  is  evidently  equal  to  the  average  length  of  the 
projection  of  the  diffusion  path  on  to  an  axis  in  the  plane 
of  observation  parallel  to  the  direction  of  the  motion  given 
to  the  solution.  The  magnitude  of  this  projection,  it  should 
be  noticed,  depends  on  the  magnitude  of  the  motion  of  the 
solution.  If  nm  denote  the  number  of  maxima  in  the  curve 
passed  over  by  the  particle  in  one  second,  and  V8  the  velocity 
given  to  the  solution,  we  have  the  relation 


Each  of  the  quantities  in  the  foregoing  equation  may  be 
measured  directly.  The  length  of  path  lp  traced  out  in  the 
plane  of  observation  in  one  second  may  also  be  obtained 
by  direct  measurement  from  the  curve.  This  length  lp 
is  equal  to  the  length  of  the  curve  that  would  be  traced 
out  in  the  plane  of  observation  if  the  particle  passed  over 


DIRECT  DETERMINATION  OF  DIFFUSION  PATH       193 

its  diffusion  path  according  to  its  definition  instead  of  over 
its  actual  path.  The  value  of  lp/nm  is  thus  equal  to  the 
length  of  the  average  projection  of  the  diffusion  path  on 
to  the  plane  of  observation  under  the  conditions  of  the 
experiment.  Therefore,  since  the  projection  of  the  dif- 
fusion path  on  to  an  axis  parallel  to  the  motion  of  the  fluid 
in  the  plane  of  observation  is  X,  it  follows  from  geometrical 
considerations  that  the  projection  of  the  path  on  to  an  axis 
in  the  plane  of  observation  at  right  angles  to  the  motion 
of  the  solution  is 

or 


since  nm  =  T,/X.  This  projection  is  independent  of  the  motion 
given  to  the  solution,  and  is  therefore  equal  to  p  the  pro- 
jection corresponding  to  the  solution  at  rest.  Therefore, 
since  we  have  previously  obtained  that  p  =  ls/2,  we  have 


(140) 


Thus  we  see  that  it  is  possible  to  determine  directly  the 
average  diffusion  path  of  a  colloidal  particle  as  defined 
in  Section  41. 

It  is  possible  therefore  to  calculate  the  coefficient  of 
diffusion  and  the  total  average  velocity  of  such  a  particle. 
According  to  equation  (129)  the  coefficient  of  diffusion 
Dr  of  particles  r  in  a  substance  consisting  of  particles  e 
is  given  by 


''    Nr' 

where  nr  denotes  the  number  of  particles  r  crossing  a  square 
cm.  from  one  side  to  the  other  per  second,  and  Nr  the  con- 


194    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

centration  of  the  particles.      Now  according  to  equation 
(35). 

_VtrNr 
Mr  o      > 

while  directly 


where  tr  denotes  the  average  period  of  the  diffusion  path 
lsr  of  a  particle  r,  and  Vtr  its  total  average  velocity.  The 
diffusion  equation  may  therefore  be  written 

D,=g.      ......    (141) 

The  period  tr  is  also  given  by 


and  may  thus  be  determined  from  curves  of  the  nature 
shown  in  Fig.  17.  Equation  (141)  may  therefore  be  used 
to  determine  the  coefficient  of  diffusion  of  colloidal  particles. 
Since  I  have  not  previously  published  the  definition 
of  the  free  diffusion  path  used,  and  the  method  of  deter- 
mining it  directly  in  the  case  of  a  colloidal  particle,  no 
values  of  it  have  yet  been  determined.  But  the  average 
values  of  the  amplitudes  A  i  (Fig.  17)  of  the  curve  described 
by  a  particle  in  the  plane  of  observation  are,  however,' 
available,  which,  we  have  seen,  are  approximately  equal 
to  l6/4.  Svedberg  has  measured  the  average  amplitude 
AI,  and  average  period  t,  of  platinum  particles  in  various 
solvents.  Using  these  values  I  have  calculated  the  co- 
efficients of  diffusion  of  the  particles,  which  will  be  found 
in  Table  XXII.  These  values  are  probably  nearer  'to  the 
truth  than  could  be  obtained  by  direct  measurement. 
The  last  column  in  the  Table  gives  the  product  Drj,  where 


EQUATION  FOR  TOTAL  AVERAGE  VELOCITY       195 

?7  denotes  the  viscosity  of  the  solvent.  It  will  be  seen  that 
it  is  very  approximately  constant,  and  the  coefficient  of 
diffusion  of  a  particle  thus  varies  inversely  as  the  viscosity 
of  the  medium.  This  is  what  we  would  expect  to  hold  in 
the  case  of  particles  considerably  larger  than  a  molecule. 

TABLE  XXII 


Solvent. 

Radius 
in 
cms.  X 

105. 

Temp. 
Cent. 

Time  t 
in 
seconds. 

Viscos- 
ity ij. 

4  A,  in 
cms.  X 

105. 

DX10' 
cm.2 
sec. 

Di)  10" 

Acetone  
E  Acetate 

0.25 
0.25 

18 
19 

0.032 
0.028 

.0023 
.0046 

14.2 
9.4 

2.10 
1.05 

4.83 
483 

Amyl  acetate  .  . 
Water  
Propyl  alcohol  . 

0.25 
0.25 
0.25 

18 
20 
20 

0.026 
0.013 
0.009 

.0059 
.0102 
.0226 

8.0 
4.3 
2.4 

.82 
.47 
.21 

4.84 
4.80 
4.75 

An  expression  for  the  total  average  velocity  of  a  colloidal 
particle  is  obtained  from  the  two  preceding  equations  giv- 
ing the  period  of  the  diffusion  path,  and  equation  (140), 
which  gives 

Vtr  =  l-^s  =  2\/lp2-Vs2.  (142) 

A 

Since  values  of  lp  are  not  available  this  equation  cannot 
yet  be  used  to  calculate  values  -of  Vtr.  We  may,  how- 
ever, use  the  approximate  average  value  of  4Ai  for  1ST, 
which  gives  to  the  equation  the  form 


This  equation  would  hold  exactly  if  the  actual  path  of  a 
particle  were  along  a  number  of  straight  lines  joined  to- 
gether at  various  angles.  It  has  been  obtained  by  Sved- 
berg  on  this  supposition,  and  used  to  calculate  the  veloci- 
ties of  particles.  For  the  platinum  particles  in  various 


196    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

solvents  to  which  Table  XXII  refers  he  obtained  the  veloci- 
ties given  in  Table  XXVI.  It  will  be  seen  that  the  velocities 
do  not  differ  much  from  one  another,  and  therefore  do  not 
seem  to  depend  much  on  the  nature  of  the  solvent  since  the 
particles  have  the  same  diameter.  The  values  obtained 
are,  however,  not  exact,  as  explained  before. 

We  have  seen  in  Section  42  and  this  Section  that  the 
coefficient  of  diffusion  depends  directly  on  the  total  average 
velocity  of  a  molecule  and  the  nature  of  its  motion.  Since 
this  also  holds  for  other  quantities,  the  coefficients  may  also 
be  expressed  in  terms  of  them.  This  applies  to  the  quan- 
tities osmotic  pressure  and  coefficient  of  mobility,  and 
their  relation  to  the  coefficient  of  diffusion  will  therefore 
now  be  investigated. 

45.  The  Coefficient  of  Diffusion  in  Connection 
with  Osmotic  Pressure  and  the  Coefficient  of  Molec- 
ular Mobility. 

Consider  a  heterogeneous  solution  of  molecules  e  and  r, 
in  which  case  diffusion  of  molecules  from  one  place  to 
another  takes  place  until  the  molecules  are  uniformly  dis- 
tributed. If  a  semipermeable  membrane  impervious  to — 
say  molecules  r,  were  placed  at  right  angles  to  a  stream  of 
diffusing  molecules  r,  they  would  exert  a  pressure  in  the 
direction  of  their  migration  upon  the  membrane.  .  Since 
the  concentration  of  the  molecules  r  is  different  on  the 
two  sides  of  the  membrane,  this  pressure  is  the  difference 
between  the  osmotic  pressures  of  the  molecules  acting  on  the 
two  sides  of  the  membrane.  This  pressure  is  therefore 
the  force  acting  upon  the  molecules  tending  to  move  them 
to  places  of  lower  concentration,  which  manifests  itself  as 
a  pressure  by  reaction  on  the  membrane  placed  in  the 
path  of  the  molecules  to  prevent  their  migration.  In  the 
absence  of  the  membrane  this  force  is  spent  in  overcoming 


CONNECTION  OF  DIFFUSION  AND  MOBILITY       197 

the  viscous  friction  exerted  by  the  mixture  on  the  migrat- 
ing molecules.  We  may  therefore  look  upon  each  cubic  cm. 
of  molecules  r  as  being  under  a  force  equal  to  the  difference 
in  the  osmotic  pressures  of  the  opposite  faces  of  the  cubic 
cm.,  which  force  is  spent  in  giving  motion  to  the  molecules 
,  against  the  viscous  friction  of  the  medium.  This  idea  of 
looking  upon  diffusion  is  mainly  due  to  Nernst. 

In  general  if  5r  denotes  the  number  of  molecules  diffusing 
across  a  square  cm.  per  second  we  have 

8r  =  NrV'r, 

where  Nr  denotes  the  concentration  of  the  molecules  r, 
and  V'r  the  velocity  with  which  each  molecule  r  on  the  aver- 
age is  moving  in  the  direction  of  decrease  of  concentration 
gradient.  Now  according  to  the  foregoing  considerations  we 
may  write 

_MrdPsr 
T     Nr   dx  ' 

where  Mr  denotes  the  coefficient  of  mobility  of  a  molecule, 
or  its  velocity  under  the  action  of  unit  force,  and  dPsr/dx 
denotes  the  osmotic  pressure  gradient.  Hence  the  preceding 
equation  may  be  written 


(143) 


Similarly  in  the  case  of  the  molecules  e  diffusing  in  the 
opposite  direction  we  have 


(144) 


From    the    preceding    equations    and    equation    (127) 
we  have, 

f-.Uf,,      ....     (145) 


198     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

an  equation  which  expresses  the  relation  between  the 
osmotic  pressure  gradients  of  the  molecules  r  and  c,  their 
coefficients  of  mobility,  and  their  molecular  volumes. 

In  the  case  of  a  dilute  solution  —  say  of  molecules  r  in 
e,  the  osmotic  pressure  obeys  the  gas  laws,  that  is  we  may 
write 


mar 


where  mr  and  mar  denote  the  relative  and  absolute  molecular 
weights  of  a  molecule  r,  and  the  ratio  mr/mar  is  therefore 
equal  to  the  number  N  of  molecules  in  a  gram  molecule 
which  according  to  Section  3  is  equal  to  6.2X1023.  Hence 


dx  ~~N~"dx' 
and  equation  (143)  becomes 


This  gives  for  the  coefficient  of  diffusion 

......     (147) 

Thus  if  the  value  of  either  of  the  two  quantities  Dr  and  Mr 
be  known  that  of  the  other  quantity  may  be  immediately 
calculated.  Table  XXIII  gives  the  values  of  M,  or  the 
mobility  under  unit  force,  of  a  number  of  different  mole- 
cules in  different  liquid  and  gaseous  media,  calculated 
from  the  known  values  of  D.  Table  XXIV  gives  the  cal- 
culated values  of  D  for  some  electrically  charged  mole- 
cules whose  mobilities  per  unit  force  are  known.  They 
agree  fairly  well  with  the  values  obtained  by  experiment. 


COEFFICIENTS  OF  MOBILITY  AND  DIFFUSION      199 


TABLE  XXIII 


Gaseous 

Gaseous 

medium 

medium 

Molecules 

of  CO2  at 
760  cm. 

of  H2  at 
760  cm. 

Liquid  medium  of  water. 

for  which 
D  is  given 
in  Table 

pressure 
and  O°  C. 

pressure 
and  O°  C. 

XX. 

I 

M  10  -12 

M  10-12 

Molecules. 

Temp. 
C. 

,    -,    2 

M  10-9. 

H2O 

*  3.6 

18.8 

Hydrogen. 

10 

3.75 

1.14 

CH40 

2.4 

13.7 

f   Nitrous 
I    oxide. 

}l4 

.63 

.19 

C6H6 
C9H1802 

1.99 
.83 

8.02 
4.70 

f     Cane 
I    Sugar. 

}l2 

.284 

.087 

TABLE  XXIV 


Nature  of 
gas  in 
which  the 
ions  are  ' 

Values  of  M  for 
+  and  —  ions 
calculated  from 
experimental 
results  obtained 
by  Wellisch.* 

Values  of  D  for 
+  and  —  ions 
calculated  by 
means  of  equa- 
tion (147). 

Values  of  D 
obtained  di- 
rectly by  ex- 
periment by 
Townsend.f 

produced. 

M-  1012 

M+  lO12 

D-. 

D  +  . 

D- 

D  +  . 

H2 

5.13 

4.32 

.200 

.169 

.19C 

.123 

O2 

1.16 

.88 

.0453 

.0343 

.0396 

.025 

C02 

.55 

.52 

.0214 

.0201 

.026 

.023 

*Phil.  Trans.;  A,  Vol.  CXCIII,  p.  129  (1900) 
t Ibid.,  A,  Vol.  CCIX,  p.  269  (1909). 


On  equating  the  coefficient  of  diffusion  given  by  equa- 
tion  (146)   with  that  given  by  equation   (129),  we  have 


Since  nr  = 


VtrNr 


(Section  18),  and  Vtrtr  =  lSr,  where  tr  denotes 


200     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

the  time  the  molecule  passes  over  the  path  I8r  with  the  velocity 
Vtr  (Section  17),  the  foregoing  equation  may  be  written 


7   2 

I'dr 

T 


RT 

rN' 


(148) 


This  equation  may  be  used  to  calculate  the  mobility  under 
unit  force  of  particles  of  a  size  that  undergo  Brownian 
motion.  The  values  of  I8r  and  tr,  we  have  seen  in  the  pre- 
vious Section,  can  be  determined  by  direct  observation. 
Table  XXV  gives  the  mobility  under  unit  force  of  particles 

TABLE  XXV 


PLATINUM  PARTICLES 

OF    AVERAGE    RADIUS 

.25X10~5  CM. 

Medium. 

Viscosity 

Mrw-*^ 

sec. 

Acetone           .            

.0023 

5.36 

E  acetate 

.0046 

2.68 

Amyl  acetate  

.0059 

2.10 

Water  
Propyl  alcohol  

.0102 
.0226 

1.20 
.54 

suspended  in  different  kinds  of  solutions  calculated  from 
data  contained  in  Table  XXII,  using  for  lSr  the  approximate 
value  4Ai. 

An  expression  may  be  found  for  the  force  F  on  a  colloidal 
particle  moving  with  the  observed  velocity  Vc  under  unit 
electric  field.  This  force  cannot  be  calculated  from  elec- 
trical data  without  the  introduction  of  assumptions.  A 
colloidal  particle  as  a  whole  is  not  electrically  charged,  but 
is  supposed  to  be  surrounded  by  two  parallel  layers  of 
electricity  of  equal  magnitude  but  of  opposite  sign.  These 
layers  get  distorted  when  an  electric  field  is  applied  to  the 
solution,  and  the  tendency  of  the  particle  to  move  so  as  to 


FORCE  ON  PARTICLE  IN  AN  ELECTRIC  FIELD      201 

readjust  this  tends  to  give  it  a  continual  motion.    The  force 
F  acting  on  the  particle  is  immediately  given  by 

Vc=FMr, 
which  may  be  written 


(149) 


by  means  of  equation  (148).  Each  quantity  on  the  right- 
hand  side  of  this  equation  may  be  determined  directly  and 
hence  F  calculated. 

Experiment  shows  that  the  observed  velocity  Vc  of  a 
colloidal  particle  varies  inversely  as  the  viscosity  of  the 
solution,  or  as  1/r/,  but  is  independent  of  the  mass  of  the 
particle.  Hence  for  different  solvents  the  force  F  is  inversely 
proportional  to  77,  and  proportional  to  tr/l2dT)  or  inversely 
proportional  to  the  mobility,  according  to  equation  (148). 
Since  the  mobility  is  proportional  to  the  coefficient  of 
diffusion  D  according  to  equation  (147),  and  Drj  is  constant 
according  to  Table  XXII,  it  follows  that  the  force  acting 
on  a  colloidal  particle  in  a  solution  under  unit  electric  field 
is  approximately  independent  of  all  conditions  at  constant 
temperature. 

It  will  be  of  interest  to  determine  the  absolute  value  of 
F  in  a  special  case.  Thus  Bredig  found  that  Vc  in  the 
case  of  platinum  particles  suspended  in  water  had  a  value 
of  about  .00025  cm.  per  second  for  an  electric  field  of  one 
volt  per  cm.  According  to  Table  XXII  for  platinum 
particles  of  radius  .25X10"5  cm.  suspended  in  water,  tr  = 
.013  sec.,  4Ai  =  .000043  cm.  corresponding  to  T  =  293°, 
while  #  =  8.315X107,  and  N  =  6.2X1023.  Assuming  that 
=  Z5r,  which  holds  approximately,  we  obtain  that 

F  =  2XlO-10dyne 


.  202     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

for  a  platinum  particle  under  the  foregoing  conditions. 
This  is  the  force  that  would  act  on  the  particle  if  it  possessed 
an  electric  charge  equal  approximately  to  lOOe. 

The  formulae  for  the  diffusion  and  viscosity  given  in  Sec- 
tions 42  and  35  involve  the  quantities  nr  and  ne,  the 
number  of  molecules  r  and  e  crossing  respectively  a  square 
cm.  from  one  side  to  the  other  per  second.  These  quan- 
tities may  be  expressed  in  terms  of  quantities  which  have 
not  been  denned  previously. 

46.  Partial  Intrinsic  Pressures.* 

In  connection  with  the  intrinsic  pressures  of  a  mixture 
it  will  be  convenient  to  introduce  the  quantity  partial  in- 
trinsic pressure,  whereby  expressions  for  various  quantities 
are  furnished  which  are  of  theoretical  interest  and  often  of 
practical  use.  Let  us  consider  a  mixture  of  molecules 
r  and  e  cut  into  two  parts  by  an  imaginary  plane  ab  as 
shown  in  Fig.  7.  Let  Pnn  denote  the  attraction  which 
the  molecules  r  in  the  portion  B  exert  on  the  molecules 
r  in  a  cylinder  of  unit  cross-section  and  infinite  length 
standing  in  the  portion  A  with  one  of  its  bases  on  the  plane 
ab,  and  let  Png2  have  a  similar  meaning  in  reference  to  the 
molecules  e.  Let  Pnre  denote  the  attraction  of  the  mole- 
cules r  in  the  portion  B  on  the  molecules  e  in  the  cylinder, 
which  is  also  equal  to  the  attraction  of  the  molecules  e 
in  this  portion  on  the  molecules  r  in  the  cylinder.  The 
intrinsic  pressure  of  the  mixture  in  the  plane  ab,  which  is 
the  sum  of  the  foregoing  forces  of  attraction,  is  therefore 
given  by 

,        ....       (150) 


where  the  quantities  on  the  right-hand  side  of  the  equation 

may  be  called  the  partial  intrinsic  pressures  of  the  mixture. 

*  Matter  published  for  the  first  time. 


PARTIAL  EXPANSION  PRESSURES  203 

The  molecules  r  in  one  of  the  portions  of  the  mixture 
are  thus  attracted  by  the  other  portion  as  a  whole  by  a 
force  which  gives  rise  to  the  pressure  Pnn+Pnre  per  cm.2 
in  the  plane  ab.  This  force  acting  on  the  molecules  r  can 
be  sustained  only  by  their  expansion  pressure,  in  other 
words,  no  part  of  the  force  can  be  sustained  by  the  expan- 
sion pressure  of  the  molecules  e  in  the  plane  ab.  For  suppose 
the  part  IT  of  the  intrinsic  pressure  Pnn+Pnre  is  sustained 
by  the  part  Xe  of  the  expansion  pressure  of  the  molecules  e. 
The  molecules  r  may  then  be  said  to  be  under  a  component 
force  IT,  and  the  molecules  e  under  a  component  force 
Xe,  acting  in  the  opposite  directions,  which  react  upon 
each  other  through  the  medium  of  the  interaction  of  the  mole- 
cules r  and  e  of  the  mixture.  But  if  two  forces  act  upon 
two  different  sets  of  molecules  in  this  way  they  would  tend 
to  be  set  in  motion  in  opposite  directions,  and  would  react 
upon  each  other  only  through  the  viscous  resistance  they 
would  exert  upon  each  other.  The  two  sets  of  molecules 
would  therefore  gradually  get  separated  through  diffusing 
through  each  other.  But  this  does  not  take  place  since 
the  mixture  is  in  equilibrium.  The  forces  in  question  are 
therefore  equal  to  zero,  or  the  intrinsic  pressure  Pnn+Pnre 
associated  with  the  molecules  r  is  sustained  by  a  part  of 
their  expansion  pressure. 

A  quantity  similar  to  the  partial  intrinsic  pressure  may 
be  denned  in  connection  with  the  external  pressure  of  a 
mixture.  This  pressure  is  exerted  in  part  by  each  set  of 
molecules,  and  in  the  case  of  a  mixture  of  molecules  e  and 
r  the  total  external  pressure  p  may  therefore  be  written 


(151) 


where  pr  and  pe  denote  respectively  the  partial  external 
pressures  of  the  molecules  r  and  c.  It  is  obvious  that  the 
partial  external  pressure  pT  represents  a  part  of  the  expan- 


204     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

sion  pressure  of  the  molecules  r,  and  a  similar  remark  applies 
to  the  partial  external  pressure  of  the  molecules  e.  It 
follows  then  from  Section  20  that  if  nr  denote  the  number 
of  molecules  r  crossing  a  square  cm.  in  the  mixture  from  one 
side  to  the  other,  and  ne  has  a  similar  meaning  with  respect 
to  the  molecules  e,  we  have 


Pr  ~        nr*  nre  =  nT 


=  2.1 

Vr  — 

and 

Ve        Ae 


e 
Vg  ~"~  0  j    £ 

=  2.543  X  lO-20^—  V  VT^e.     (153) 


The  partial  intrinsic  pressures  cannot  yet  be  expressed 
similarly  as  the  quantity  Pn  in  terms  of  other  quantities 
which  can  be  directly  measured.  Approximate  numerical 
values  may,  however,  be  obtained  from  the  simultaneous 
equations  (150),  (152),  and  (153),  by  neglecting  the  partial 
external  pressures  in  comparison  with  the  partial  intrinsic 
pressures,  which  may  usually  be  done,  and  calculating  the 
quantities  nr,  ne,  b'r,  b'e,  and  Pn,  by  the  method  of  Sections 
21  and  29. 

Approximate  values  may  also  be  obtained  by  the  fol- 
lowing method.  In  the  case  of  a  pure  substance  the  intrinsic 
pressure  according  to  Section  26  may  as  a  first  approxima- 
tion be  taken  proportional  to  the  square  of  the  density  p 
of  the  substance.  Applying  this  result  to  the  kind  of  mixture 
under  consideration  the  quantities  Pnr*  and  Pne2  may  evi- 
dently be  written 

Pnf,  =  JBVr,   .     .     o     .     .     .     (154) 
and 

.....       (155) 


THE  INTRINSIC  PRESSURE  AND  LATENT  HEAT    203 

where  pr  and  pe  denote  the  partial  densities  of  the  molecules 
r  and  e  in  the  mixture,  and  BT  and  Be  denote  approximate 
constants.  As  a  first  approximation  the  quantity  PnTe  is 
then  given  by 

.....       (156) 


The  values  of  the  quantities  Br  and  Be  may  be  obtained 
from  the  internal  heats  of  evaporation  Lr  and  Le  of  gram 
molecules  of  the  substances  in  the  pure  state.  Thus  accord- 
ing to  Section  21  we  have 


Lr  = 


where  pi  and  p2  denote  the  densities  of  the  pure  substance 
r  in  the  liquid  and  vaporous  states,  and  a  similar  equation 
may  be  obtained  for  Le.  These  equations  express  Br  and 
Be  in  terms  of  Lr  and  Le. 

The  values  for  the  partial  intrinsic  pressures  obtained 
by  this  or  the  preceding  method  may  be  tested  by  sub- 
stituting them  in  equation  (150),  and  determining  Pn 
(the  intrinsic  pressure  of  the  mixture  as  a  whole)  by  the 
method  described  in  Section  21.  If  an  agreement  is  obtained 
we  may  be  fairly  sure  that  the  values  of  the  partial  intrinsic 
pressures  obtained  are  very  approximately  correct. 

If  the  latter  method  is  used  to  determine  the  partial 
intrinsic  pressures,  we  may  use  equations  (152)  and  (153) 
to  determine  ne  and  nr.  We  are  thus  furnished  with  another 
method  of  determining  the  latter  quantities,  which  is  com- 
paratively simpler  to  use  than  that  described  in  Section  29. 

The  latter  method  of  obtaining  the  partial  intrinsic 
pressures  may  evidently  also  be  used  to  find  the  intrinsic 
pressure  of  a  pure  substance. 

In  the  case  of  a  mixture  of  more  substances  than  two 
the  corresponding  partial  intrinsic  pressures  will  not  be 


206    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

difficult  to  define,  and  may  be  obtained  in  a  similar  way  as 
described. 

The  partial  intrinsic  pressures  of  a  heterogeneous  mixture 
may  also  be  connected  with  its  osmotic  pressures.  Before 
discussing  the  connection  it  will  be  necessary  to  consider 
some  general  conditions  of  equilibrium. 

47.  Conditions  of  the  Equilibrium  of  a  Hetero- 
geneous Mixture  such  as  Two  Phases  in  Contact. 

The  relative  concentration  of  the  constituents  of  two 
phases  of  a  mixture  is  usually  different  and  gradually  changes 
from  one  to  the  other  in  the  transition  layer  which  exists 
at  the  boundary  of  the  phases.  In  the  case  of  a  mixture 
of  molecules  r  and  e,  for  example,  equations  (152)  and  (153) 
must  hold  for  both  phases  and  the  transition  layer,  and  are 
therefore  two  of  the  equations  of  equilibrium. 

The  partial  external  pressures  pr  and  pe  in  these  equa- 
tions must  have  the  same  values  everywhere,  otherwise 
diffusion  from  one  portion  to  the  other  would  take  place,  and 
the  system  would  not  be  in  equilibrium.  This  follows  at 
once  from  considering  two  layers  of  liquid  in  which  the 
partial  external  pressures  have  the  values  pr,  pe,  and  p'r, 
p'e,  respectively.  Since  the  total  pressure  is  the  same 
everywhere 

Pe  +  Pr  =  p'e  +  P'r, 

and  hence 


From  the  latter  equation  it  follows  that  there  is  an  excess 
of  pressure  of  the  molecules  r  in  one  direction  which  is 
balanced  by  an  excess  of  pressure  of  the  molecules  e  in  the 
opposite  direction.  But  this  would  give  rise  to  a  diffusion 
of  the  two  sets  of  molecules  through  each  other,  reasoning 


EFFECT  OF  INTRINSIC  PRESSURE  GRADIENT     207 

along  the  same  lines  as  in  the  previous  Section,  and  thus 
disturb  the  equilibrium.    Hence  we  must  have 

Pr  =  p'r  and  pe=p'e  .....        (157) 

If  nr  denotes  the  number  of  molecules  r  crossing  a  square 
cm.  from  one  side  to  the  other,  n'r  the  number  crossing  in 
the  opposite  direction,  and  ne  and  n'e  have  similar  meanings 
with  respect  to  the  molecules  e,  we  must  also  have 

n'e  =  ne  } 
and  L    ......     (158) 


otherwise  diffusion  from  one  place  to  the  other  would  take 
place. 

The  foregoing  equations  are  of  special  interest  in  con- 
nection with  the  transition  layer,  since  this  has  molecular 
concentration  gradients  the  tendency  of  which  is  to  give 
to  nr,  the  molecules  crossing  a  square  cm.  in  one  direction, 
a  value  different  from  n'r,  the  number  crossing  in  the  opposite 
direction,  and  to  give  also  different  values  to  ne  and  n'e. 
But  there  evidently  exists  a  force  in  the  layer  at  right  angles 
to  it  acting  in  the  direction  of  increase  of  density,  due  to 
the  existence  of  molecular  forces  of  attraction.  This  force 
is  equal  to  the  intrinsic  pressure  gradient.  It  has  the  effect 
of  decreasing  the  velocity  of  the  molecules  moving  in  the 
direction  of  decrease  of  density  and  increasing  the  velocity 
of  the  molecules  moving  in  the  opposite  direction.  The 
magnitude  of  this  modification  of  the  velocity  of  a  molecule 
depends  on  its  nature,  since  the  various  molecules  do  not 
possess  the  same  forces  of  attraction.  Thus  the  effect  of 
the  intrinsic  pressure  gradient  on  the  number  of  molecules 
crossing  a  square  cm.  from  one  side  to  the  other  per  second 
is  opposite  to  that  of  the  concentration  gradient,  and  there- 
fore nr  and  n'r,  and  ne  and  n'e,  may  be  rendered  equal,  as  is 


.- 


208    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

the  case  in  practice.  Besides,  since  the  effect  of  the  intrinsic 
pressure  gradient  on  a  set  of  molecules  depends  on  their 
nature,  the  concentration  gradient  would  not  necessarily 
be  the  same  for  each  set  of  molecules  corresponding  to  the 
equilibrium  conditions  (152)  and  (153).  Thus  we  see  why 
the  relative  concentration  of  the  ingredients  of  a  mixture 
differs  in  the  vaporous  phase  from  that  in  the  liquid  phase. 
It  will  also  be  evident  that,  since  the  attraction  of  a  molecule 
increases  with  its  mass,  the  relative  concentration  of  the 
heavier  molecule  is  likely  to  be  smaller  in  the  vaporous 
phase  than  in  the  liquid  phase.  This  is  exemplified  in  prac- 
tice, though  it  need  not,  and  does  not,  hold  in  every  case. 

Similar  conditions  apply  to  a  mixture  of  more  than  two 
constituents. 


48.  Osmotic  Pressure  expressed  in  Terms  of 
the  Kinetic  Properties  of  Molecules.* 

If  the  different  portions  of  a  mixture  of  substances  are 
not  in  equilibrium  with  each  other,  which  would  manifest 
itself  by  a  redistribution  of  the  ingredients  by  diffusion, 
the  equations  of  equilibrium  given  in  the  foregoing  Section 
will  not  hold.  Evidently  equations  (157)  will  not  hold  since 
the  vapor  pressure  of  the  molecules  r  depends  on  their 

concentrations.    A  force  equal  to  —  ,  where  x  is  measured 

ox 

along  a  line  of  decrease  of  concentration,  will  therefore  act 
on  each  cubic  cm.  of  molecules  r.  Equation  (152)  may  hold 
at  certain  parts  of  the  mixture,  though  in  general  this  would 
not  be  the  case.  There  is  therefore  an  additional  force 
equal  to 


Matter  published  for  the  first  time. 


THE  NATURE  OF  OSMOTIC  PRESSURE  209 

acting  on  the  molecules  r  per  cubic  cm.     The  total  force  Fr 
acting  on  the  molecules  r  per  cubic  cm.  is  therefore  given  by 

p  —  _i_  \-^L     VT     $Hi-L-^L     HrVr      dfr'r 

tl  2    *-*V    .ifeT.2    (Vr-b'r)2^ 

<Arnrf        1  Vr 

T      2 


'.    .    (159) 

This  force  is  equal  to  the  osmotic  pressure  acting  on  the 
molecules  r.  It  gives  rise  to  a  diffusion  of  molecules  equal 
to  nr—  n'r'y  and  equations  (158)  therefore  also  do  not  hold. 
It  is  likely  that  the  upper  sign  of  the  first  term  of  the  right- 
hand  side  of  the  foregoing  equation  always  holds  in  practice. 
Similarly  the  force  Fr  acting  on  the  molecules  e  per 
cubic  cm.  giving  rise  to  a  diffusion  of  molecules  e  in  the  oppo- 
site direction  to  the  molecules  r,  is  given  by 

\Ae        Ve         faeAe         UeVe         db'e 


2      V- 


,Aene/     1  ve      \  8ve 

2   \ve-b'e     (Ve-b'e)2/5x 

.       .       (160) 


Since  nr  depends  on  the  motion  of  translation  of  the 
molecules  and  molecular  attraction,  the  foregoing  equations 
express  osmotic  pressure  in  terms  of  molecular  motion, 
molecular  attraction,  and  the  apparent  molecular  volume. 
It  will  be  evident  on  reflection  that  osmotic  pressure  must 
arise  through  these  properties  of  matter,  and  the  equations 
are  therefore  fundamental  in  character.  They  are,  however, 
of  little  use  in  practice  to  calculate  osmotic  pressure,  since 
we  have  no  means  yet  of  determining  experimentally  how 


210    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

the  quantities  nr,  b'r,  Pnr2,  and  Pner  vary  in  a  heterogeneous 
mixture  from  one  part  to  another. 

Osmotic  pressure  as  exhibited  in  connection  with  semi- 
permeable  membranes  will  be  discussed  in  connection  with 
another  subject. 

Since  the  quantities  ne  and  nT  in  the  foregoing  and 
other  formulae  depend  on  the  total  average  velocities  of  the 
molecules  e  and  r,  it  will  be  of  importance  to  see  if  some 
direct  information  can  be  obtained  about  the  values  of  these 
velocities. 

49.  A  Method  of  Obtaining  the  Velocity  of  Trans- 
lation of  Particles  undergoing  Brownian  Motion. 

The  path  of  such  a  particle  we  have  seen  in  Section  44, 
is  zigzag  in  shape.  The  average  velocity  of  translation  of 
the  particle  may  therefore,  according  to  this  Section,  be 
determined  from  observations  of  the  average  period  t  of  the 
free  path  ls  by  means  of  the  equation 


(161) 


Table  XXVI  contains  the  approximate  velocities  of  plati- 
num particles  in  different  solvents  determined  in  this  way 
from  observations  of  the  amplitudes  A\  contained  in  Table 
XXII  made  by  Svedberg  and  described  in  Section  44, 
which  we  have  seen  are  approximately  equal  to  Za/4.  It  is 
usually  supposed  that  the  average  velocity  of  a  colloidal  par- 
ticle is  the  same  it  would  have  in  the  perfectly  gaseous  state. 
But  on  calculating  the  average  velocity  Va  under  these  con- 
ditions, which  is  given  by  Va=  -922F  and  equation  (8),  it  is 
found  that  it  does  not  agree  with  that  observed,  as  will 
appear  from  Table  XXVI,  which  gives  the  velocities  of 
platinum  particles  obtained  in  these  two  ways.  Now  this 
is  what  we  would  expect  from  the  results  of  Sections  16  and 


THE  MOTION  OF  COLLOIDAL  PARTICLES 


211 


29.  But  it  has  yet  to  be  shown  why  the  actual  velocities 
might  be  smaller  in  the  case  of  a  colloidal  particle  than 
corresponding  to  the  gaseous  state,  when  in  the  case  of  a 
single  molecule  in  a  liquid  we  have  seen  it  has  a  greater 
value.  It  appeared  from  Section  44  that  a  colloidal  particle 
is  continually  under  the  influence  of  the  surrounding  mole- 
cules in  varying  degrees,  and  continually  bombarded  from 
all  sides  by  molecules,  whose  resultant  effect  is  not  zero 


TABLE  XXVI 


PLATINUM  PARTICLES  < 

Va  FOR   GAS 

3F    AVERAGE    RADIUS    .25X10"5    CM. 
CM. 

SEC. 

Medium. 

Viscosity. 

Vt  10»  -551: 

sec. 

Acetone                 

.0023 
.0046 
.0059 
.0102 
.0226 

444 
336 
308 
324 
266 

E  acetate 

Amyl  acetate  

Water                    

Propyl  alcohol  

but  continually  changes  in  direction  and  magnitude,  giving 
the  particle  the  well-known  zigzag  motion.  Therefore 
when  the  particle  begins  to  move  in  any  new  direction  its 
motion  is  at  once  impeded  by  the  viscous  friction  of  the 
surrounding  medium,  due  to  the  particle  being  constantly 
more  or  less  under  the  influence  of  the  surrounding  mole- 
cules. Hence  if  the  initial  velocity  is  that  corresponding 
to  the  gaseous  state,  or  greater,  it  may  be  slowed  down  to 
a  much  smaller  value  before  the  particle  receives  a  new 
impetus,  and  the  total  average  velocity  is  therefore  not 
likely  to  be  the  same  as  in  the  gaseous  state,  but  may  be 
much  less.  Section  60  also  deals  with  this  point. 


212     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

It  is  often  assumed  that  the  velocity  of  a  colloidal 
particle  obtained  from  observations  of  its  path  does  not 
represent  the  true  velocity,  since  the  amplitude  of  the 
oscillatory  motion  may  occasionally  be  so  small  that  it 
cannot  be  observed  by  means  of  the  ultra-microscope, 
and  what  in  truth  is  a  wavy  curve  of  small  amplitude  and 
wave  lengths  would  be  taken  for  a  smooth  line.  This  is 
no  doubt  partly  true.  But  it  is  highly  improbable  that 
only  TV  of  the  path  of  the  particle  should  be  possible  of 
being  observed  by  the  ultra-microscope.  Otherwise  the 
oscillatory  motion  of  the  particle  would  largely  correspond 
to  amplitudes  whose  magnitude  could  not  be  determined 
by  the  microscope,  on  which  are  impressed  amplitudes  of 
very  much  greater  magnitude.  This  does  not  seem  to  be 
in  harmony  with  the  distribution  of  the  amplitudes  according 
to  Clausius'  law  given  in  Section  31.  For  this  reason,  and 
that  the  particle  must  always  be  subject  to  viscous  fric- 
tion in  its  motion,  we  conclude  that  the  velocity  need  not 
be  that  corresponding  to  the  gaseous  state,  but  is  probably 
much  less. 

Some  experiments  by  Exner,  who  showed  that  particles 
of  diameters  1.3ju,  .Q/z,  and  .4ju,  have  the  velocities  2.7, 
3.3,  and  3.8,  respectively,  point  to  the  same  conclusion. 
For  the  masses  of  the  particles  are  proportional  to  the 
cubes  of  their  diameters,  and  their  velocities  in  the  gaseous 
state  therefore  proportional  to  the  f  power  of  their 
diameters.  But  this  is  evidently  not  realized. 

If  Vi  denote  the  velocity  of  a  particle  at  any  instant 
in  passing  over  the  distance  51,  and  R  the  viscous  resistance 
of  the  medium,  we  have 


where  ma  denotes  the  mass  of  the  particle.    The  resistance 
R  is  proportional  to  Vi,  or  equal  to  K^V\^  where  K\  denotes 


THE  MOTION  OF  A  PARTICLE  UNDER  IMPETUS    213 

a  constant,  and  hence  the  preceding  equation  on  substitut- 
ing for  R  and  integrating  gives 

l=~(V^Ve), 

K2 

where  Vt  denotes  the  initial  velocity  of  the  particle  when 
it  changes  its  direction,  Ve  the  velocity  when  it  is  about 
to  receive  a  new  impetus,  and  I  the  length  of  path  between 
the  corresponding  points. 

It  follows  therefore  that  the  rate  of  change  of  V\,  or  the 
negative  acceleration,  is  constant  when  the  particle  is  not 
propelled  by  the  molecules  of  the  medium. 

The  value  of  Kz  is  probably  very  different  from  the  value 
of  the  coefficient  of  mobility  referring  to  the  average  velocity 
of  a  particle  parallel  to  a  given  direction  considered  over  a 
time  that  gives  a  constant  velocity  under  the  action  of  an 
external  force.  The  latter  constant  has  been  considered  in 
Section  45  and  is  the  factor  of  Vc  in  equation  (149). 

The  result  of  this  Section  is  of  interest  in  connection 
with  the  osmotic  pressure  of  dilute  solutions  considered  in 
the  next  Section. 

50.  The  Osmotic  Pressure  of  a  Dilute  Solution 
of  Molecules,  and  their  Velocity  of  Translation.* 

Van't  Hoff  has  shown  that  it  follows  from  thermodynam- 
ics that  the  osmotic  pressure  of  a  dilute  solution  obeys  the 
gas  equation,  that  is,  we  may  write 


where  N  denotes  the  number  of  molecules  in  a  gram  mole- 
*  Not  previously  published. 


214     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

cule  of  a  substance,  and  Nr  the  concentration  of  the  solute 
molecules  r  in  the  solvent  e.  This  result  has  given  rise  to 
the  erroneous  statement  often  made  that  the  velocity  of 
translation  of  each  solute  molecule  is  the  same  as  that  it 
would  have  in  the  perfectly  gaseous  state.  It  is  obvious 
that  if  this  be  true  for  the  solute  molecules  it  must  also 
be  true  for  the  molecules  of  a  pure  liquid.  But  it  can  be 
shown  in  many  ways  that  this  deduction  cannot  hold,  though 
Van't  Hoff's  law  be  true. 

If  the  velocity  of  translation  of  a  molecule  is  always  the 
same  as  in  the  gaseous  state  the  expansion  pressure  of  a 


substance  would  be  given  by  the  expression  —  -,  according 

to  equation  (40)  in  Section  20.  Now  the  external  pressure 
of  a  substance  is  not  equal  to  this  expression,  and  therefore 
forces  must  exist  between  the  molecules  which  give  rise 
to  an  intrinsic  pressure  which,  together  with  the  external 
pressure,  balances  the  expansion  pressure,  as  expressed  by 
equation  (45).  But  if  the  molecules  exert  forces  of  attrac- 
tion and  repulsion  upon  each  other  these  will  have  the  effect 
of  changing  the  velocity  of  a  molecule  from  what  it  would 
have  in  the  gaseous  state  as  shown  in  Section  17.  According 
to  Sections  25  and  29  this  total  average  velocity  of  a  mole- 
cule in  a  liquid  is  many  times  what  it  would  have  in  the 
gaseous  state.  Another  method  used  in  Section  34  gave  the 
same  result. 

It  can  also  be  shown  that  this  result  is  a  consequence  of 
Section  48,  which  gives  us  besides  other  information.  Let 
us  suppose  that  the  vapor  pressure  of  the  solute  r  of  a  solu- 
tion is  zero,  as  is  the  case  of  a  salt  dissolved  in  water.  Since 
the  molecules  of  the  solute  in  the  solution  according  to  the 
definition  of  a  dilute  solution  are  so  far  apart  that  their 
influence  upon  each  other  is  practically  zero,  we  have  PnT2 


=  0.   According  to  Van't  Hoff's  law  Fr  =  —  =  ?~  —  along 

dx       N    dx 


VELOCITY  OF  PARTICLES  IN  DILUTE  SOLUTIONS     215 

a  concentration  gradient  in  the  mixture.     Equation  (159) 
therefore  in  this  case  becomes 

RT5NrAr     vr      8n,     Ar      nrvr      dVr 


N      dX          2    VT-b'r    5Z         2    (Vr-b'rY     dx 

r       dP 


. 


Now  if  the  velocity  of  translation  of  a  molecule  r  were  the 
same  as  in  the  gaseous  state  UT  would  be  given  by  equation 

(21)  on  adding  suffixes  r  to  the  symbols,  and  —  ^  would 

Ol\  r 

be  given  by  equations  (162)  and  (20),  in  which  case 


6NrNA/ 

dNr 

Therefore  on  dividing  equation  (163)   by  -—  ,  and  taking 

uX 

into  account  that  NTVrmar  =  mr,  it  may  be  written 

RT  _        Vr        RT       Ar 
" 


8Pnfe       RTvrf       I  Vr 

^    dNT          Nr    \Vr-b'r       (vr-b'r) 

But  the  right-hand  side  of  this  equation  is  not  identically 
equal  to  the  left-hand  side,  for  -  -  is  not  approxi- 

VT 

mately  equal  to  unity,  since  the  value  of  &/  according  to 
Section  20  is  determined  by  the  volume  of  the  molecules 
e  as  well  as  that  of  the  molecules  r  in  a  gram  molecule  of 
molecules  r,  and  the  value  of  b'r  is  therefore  not  small  in 

comparison  with  vr;   for  the  same  reason  —  ^  is  not  zero; 

oNr 

?r> 

and  -  -^  is  not  zero  since  molecular  forces  exist.     Thus 
dNr 


216     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

in  general  the  velocity  of  translation  of  the  molecules  r  would 
not  .be  equal  to  that  in  the  gaseous  state. 

We  may  also  note  the  following  considerations  in  this 
connection.  If  we  suppose  that  the  velocity  of  the  solute 
molecules  in  a  dilute  solution  is  the  same  as  that  which 
they  would  have  in  the  perfectly  gaseous  state  their  external 

r>/7T 

pressure  is  given  by  -  —n->     Now  the   apparent  volume 

Vr      0  r 

b'r  of  the  molecules  of  the  mixture,  or  the  apparent  volume 
which  the  molecules  e  and  r  appear  to  possess  in  obstructing 
the  motion  of  the  molecules  r,  is  not  small  in  comparison 
with  vr.  Its  value  is  approximately  equal  to  vrbe/ve,  where 
be  refers  to  the  solvent  e  in  the  pure  state.  According  to 
the  above  supposition  Van  der  Waals'  equation  holds,  and 
if  applied  to  the  solvent  e  in  the  pure  state  we  have 

RT 


(Section  26)  where  Pn,  the  intrinsic  pressure,  is  large  in  com- 
parison with  the  external  pressure  p,  and  large  (Section 
21)  in  comparison  with  the  pressure  p'  the  substance  would 
have  if  it  behaved  as  a  perfect  gas.  It  follows  therefore 
from  the  equation  that  be/ve  is  not  small  in  comparison  with 

r>/77 

unity.     Hence  on  substituting  vrbe/ve  for  b'r  in  -  —j^  it 

Vr      0  r 
r>/T7 

becomes  —r-  —  ,   ,   .,  and  thus  the  pressure  exerted  by  the 


solute  molecules  is  many  times  that  which  they  would  have 
in  the  perfectly  gaseous  state.  This  pressure  is  in  part 
balanced  by  the  intrinsic  pressure  of  the  mixture.  We 
cannot,  therefore,  with  any  show  of  reason,  say  that  because 
the  osmotic  pressure  of  the  solute  molecules  obeys  the  gas 
laws  their  velocity  of  translation  is  the  same  as  in  the 
gaseous  state. 


THE  CONSTANTS  OF  A  DILUTE  SOLUTION          217 

The  values  of  w,  and  Pnre  of  a  dilute  solution  are  evidently 
proportional  to  NT,  while  b'r  is  proportional  to  vr,  and  we 
may  therefore  write 


nr  =  aiNr,  b'r  =  a2vr,  and  PnTe  = 

where  ai,  a2,  and  a3  are  constants.     Equation  (163)  may 
therefore  be  written 

2RT(l-a2)     2(1-02)03 
-  ---,   •     .     •     (164) 


on  taking  into  account  that 


which  gives  a  relation  between  the  foregoing  constants. 

A  method  of  determining  directly  the  number  of  mole- 
cules in  a  gram  molecule  which  is  based  on  equation  (162) 
will  now  be  described. 

51  .  A  Direct  Determination  of  N,  the  Number  of 
Molecules  in  a  Gram  Molecule. 

Perrin*  has  determined  directly  the  value  of  N  by  a 
method  of  counting  the  number  of  particles  in  a  dilute 
colloidal  solution.  The  gravitational  attraction  of  the 
particles  tends  to  deposit  them  on  to  the  bottom  of  the 
vessel  containing  the  solution.  The  concentration  gradient 
caused  thereby  gives  rise  to  an  osmotic  pressure  acting  in 
the  opposite  direction  to  the  gravitational  attraction. 
Equilibrium  exists  when  these  forces  balance  one  another, 
which  corresponds  to  an  increase  in  the  concentration  of 
the  solution  with  increase  of  distance  from  the  surface. 

*  C.  R.,  147  (1908),  p.  530;  147  (1908),  p.  594;  Ann.  Chim.  Phys., 
8,  18  (1909),  pp.  5-174;  Bull  Soc.  Fr.  Phys.,  3  (1909),  p.  155;  Zs.  /. 
Electroch.,  15  (1909),  p.  269. 


(218    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

Since  the  osmotic  pressure  obeys  the  gas  laws  the  distribu- 
tion of  the  particles  is  similar  to  that  of  the  molecules  of  a 
gas  under  the  action  of  gravity.  The  differential  equation 
of  equilibrium  is 

F-dh=5Ps, 

where  Ps  denotes  the  osmotic  pressure  at  a  distance  h  from 
the  surface  of  the  solution,  and  F  the  force  due  to  gravity 
acting  on  the  particles  in  a  cubic  cm.  of  the  solution.  If 
va  denote  the  volume  of  a  particle,  pa  its  density,  p  the  density 
of  the  liquid,  we  have 

F  =  va(pa-p)gNc, 

where  Nc  denotes  the  concentration  of  the  particles.  For 
Ps  we  have 

P=RTN 

N     c> 

according  to  equation  (162),  and  hence  the  equation  of 
equilibrium  becomes 

Va(pa  —  p)gNc  '  8k  =  -JT^-  •  5NC, 


which  on  integration  gives 

\os,    .    .     (165) 


where  N'c  and  N"c  denote  the  concentrations  corresponding 
to  the  distances  h\  and  Ji2  from  the  surface  of  the  solution. 
The  values  of  N'c  and  N"c  corresponding  to  given  values  of 
h\  and  hz  were  found  by  counting  the  particles  in  different 
planes  in  a  solution  placed  under  the  ultra-microscope. 
Solutions  of  gamboge  and  mastic  were  used. 

The  densities  of  the  particles  in  the  solutions  were 
determined  by  two  methods.  In  one  method  it  was  taken 
the  same  as  that  of  the  substance  in  the  undivided  state, 


THE  CONSTANT  N  FROM  VAN'T  HOFF'S  LAW       219 

while  in  the  other  a  known  volume  of  the  solution  was 
evaporated  and  the  residue  weighed.  The  two  methods 
gave  concordant  results. 

The  volume  of  a  particle  was  obtained,  in  one  of  the  three 
methods  used,  by  counting  the  number  of  particles  in  a 
given  volume  of  emulsion  and,  knowing  their  weight  and 
density,  the  volume  was  immediately  obtained. 

In  the  second  method  the  emulsion  was  slightly  acidu- 
lated, which  has  the  effect  of  making  the  particles  stick 
together  in  little  strings  which  adhere  to  the  vessel's  walls. 
On  measuring  the  length  of  a  string  and  counting  the  num- 
ber of  particles  in  it,  their  radii  could  be  determined. 

The  third  method  depended  on  an  application  of  Stokes' 
law  to  the  rate  of  fall  of  the  particles  under  the  action  of 
gravity. 

These  methods  gave  concordant  results.  Thus  in  one 
case  the  three  methods  gave  for  the  same  particle  the  diam- 
eters .46/x,  .455ju,  and  .45ju.  Particles  of  mastic  and  gamboge 
in  a  solution  thus  appear  to  be  spherical  in  shape,  and 
their  motion  obeys  Stokes'  law. 

These  measurements  determine  the  various  quan- 
tities in  equation  (165)  except  N,  which  is  therefore  expressed 
in  terms  of  these  quantities.  In  this  way  Perrin  obtained 

N  =  7.05X1023, 

which  agrees  as  well  as  can  be  expected  with  the  value 
given  in  Section  3. 

It  is  very  important  to  notice  that  this  investigation 
does  not  depend  upon  the  velocity  of  translation  of  the 
particles  being  the  same  as  if  they  were  in  the  perfectly 
gaseous  state.  It  depends  merely  upon  the  osmotic  pressure 
obeying  the  gas  equation,  which  according  to  Section  50 
may  be  the  case  independent  of  the  velocity  of  translation 
of  the  particles.  This  point  is  of  importance  because  these 


220     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

experiments  are  occasionally  cited  as  proving,  on  account 

nfTI 

of  the  occurrence  of  the  factor  -^-  in  equation  (165),  that 

the  colloidal  particles  in  a  solution  have  the  same  velocity 
as  they  would  have  in  the  gaseous  state. 


52.  A  Free  Path  Formula  involving  Stokes'  Law. 

According  to  Stokes'  law  if  a  spherical  particle  of  radius 
r  is  under  the  action  of  a  force  F  in  a  medium  of  viscosity 
rj  and  density  p,  and  there  is  no  slipping  (Section  34), 
the  velocity  Vc  with  which  the  particle  moves  is  given  by 

F.- 


Since  Vc  =  MF,  where  M  denotes  the  coefficient  of  mobility, 
the  equation  may  be  written 

M  =  ^-.      .     .     .  (166) 

671T77 

On  substituting  this  expression  for  M  in   equation   (148) 
it  becomes* 

h2    RT    1 

(167) 


Since  the  value  of  ls  differs  little  from  four  times  the  aver- 
age amplitude  A\  of  a  particle  observed  in  practice  (Sec- 
tion 44)  the  equation  should  agree  approximately  with  the 
facts  provided  Stokes'  law  holds.  If  for  example  the  values 
of  4A  i  and  t  (Section  44)  observed  by  Svedberg  for  platinum 
particles  are  substituted  in  the  left-hand  side  of  the  equation 
for  ls  and  t  it  does  not  agree  even  approximately  with  the 
values  of  the  right-hand  side,  as  is  shown  in  Table  XXVII. 
This  may  be  caused  by  the  values  of  kA\  differing  more 
*  Not  previously  published. 


AN  EQUATION  OF  MOTION  OF  PARTICLES          221 

from  the  values  of  ls  than  is  apparent.  The  values  of  4Ai 
are  probably  otherwise  unobjectionable  since  the  coefficients 
of  diffusion  of  platinum  particles  in  different  solvents  cal- 
culated from  them  (Section  44)  give  very  approximately 
Dry  =  constant.  It  is  more  likely  that  the  motion  of  plati- 
num particles  in  a  solution  obtained  by  the  sparking  of 
platinum  electrodes  in  a  solvent  does  not  obey  Stokes'  law. 
This  is  perhaps  not  surprising  since  these  particles  are  not 
likely  to  be  spherical  in  shape  as  required  by  this  law, 
but  more  likely  consist  of  flakes,  since  they  are  produced 
by  portions  of  the  electrodes  being  torn  off  through  the 
electrical  discharge. 

The  particles  of  gamboge  used  by  Perrin  in  the  investi- 
gation described  in  the  previous  Section  are  more  likely 
to  be  spherical  in  shape  since  they  were  prepared  by  rub- 
bing gamboge  in  distilled  water  giving  a  yellow  solution 
containing  particles  of  various  sizes  whose  corners  would 
more  or  less  be  dissolved  off.  Thus  the  motion  of  such 
particles  might  obey  Stokes'  law,  as  Perrin  has  directly 
observed.  Values  of  ls  and  t  for  such  particles  are,  however, 
not  available  to  test  equation  (167). 

But  whatever  the  geometrical  configuration  of  the 
particle  the  velocity  Vc  is  likely  to  vary  inversely  as  r/,  or 


(168) 


where  Kc  is  a  constant  depending  on  the  nature  of  the 
particle;  and  accordingly  equation  (148),  since  VC  =  MF, 
may  be  written 

182_3RT  Kc  nfiqx 

T  nr  rj- 

This  equation  appears  to  agree  in  a  general  way  with  the 
facts.  It  might  be  used  to  calculate  Kc,  which  has  been 
carried  out  for  platinum  particles  in  Table  XXVII,  using 


222    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS  . 


the  data  contained  in  Table  XXII.  The  values  obtained 
are  practically  constant,  as  we  would  expect  that  they  should 
be  if  the  mass  of  the  particles  is  kept  the  same. 

TABLE   XXVII 


PLATINUM  PARTICLES  OF  RADIUS  .25X10    °  CMS. 

Solvent.' 

107's2/< 

107  RT        l 

D  ~N~2^, 

Kc10-« 

Acetone  

6.45 
2.51 
2.46 
1.42 
.640 

10.9 
5.43 
4.24 
2.45 
1.11 

3.688 
2.934 
3.696 
3.693 
3.681 

E.  acetate  
Amyl  acetate 

Water 

Propyl  alcohol    

Some  interesting  and  important  extensions  of  the  vis- 
cosity, conduction  of  heat,  and  diffusion  formula?  will 
now  be  given  which  involve  an  extended  meaning  of  the 
molecular  paths  under  various  conditions.  Following 
these  a  number  of  important  formulae  related  to  the  fore- 
going will  be  developed  involving  the  projection  of  the 
motion  of  a  molecule  along  an  axis,  instead  of.  the  molecular 
path.  By  means  of  these  formulae  further  information 
about  the  motion  of  a  molecule  and  other  of  its  properties 
may  be  obtained.  The  foregoing  developments  are  new, 
and  have  not  been  published  previously. 

53.  Extended  Forms  of  the  Diffusion  Equations., 

The  most  important  property  of  the  mean  diffusion 
path  of  a  molecule  defined  in  Section  41  which  is  contained 
in  its  definition  is,  that  the  sum  of  the  diffusion  paths  be- 
tween two  points  is  equal  to  the  corresponding  length  of 
the  actual  path  of  the  molecule.  This  property  enables  us 
to  express  the  number  of  diffusion  paths  n  crossing  a  plane 


AN  EXTENDED  TREATMENT  OF  DIFFUSION        223 

one  square  cm.  in  area,  a  quantity  which  occurs  in  the 
diffusion  equations,  in  terms  of  quantities  which  can  be 
determined  directly  or  indirectly.  We  may,  however,  give 
other  definitions  to  the  diffusion  path,  which,  however,  do 
not  enable  us  to  express  in  a  simple  way  the  corresponding 
value  of  n  in  terms  of  other  quantities.  Thus  we  may  take 
a  number  of  points  on  the  path  of  a  molecule  subject  to  any 
condition  we  please,  join  consecutive  points  by  straight 
lines,  and  suppose  that  the  molecule  in  its  migration  passes 
along  these  straight  lines  instead  of  along  its  actual  path. 
The  points,  it  should  be  noted,  need  not  even  lie  on  the 
actual  path  of  the  molecule.  The  straight  lines  thus  obtained 
constitute,  as  usual,  the  representative  diffusion  paths  of  a 
molecule  under  the  stated  conditions. 

Suppose  for  example  that  the  points  are  taken  on  the 
path  of  a  molecule,  and  are  so  selected  that  the  diffusion 
paths  are  grouped  about  a  mean  path  l'dr  of  any  chosen 
length  according  to  Clausius7  distribution  law  given  in 
Section  31,  and  that  any  direction  of  a  path  in  space  is 
equally  probable.  It  follows  then  from  the  investigation 
in  Section  42  that  the  expression  for  the  coefficient  of  dif- 
fusion of  a  dilute  solution  of  molecules  r  in  e  is  the  same  in 
form  as  the  coefficient  given  by  equation  (129),  or 

D,  =  ~jf, (170) 

where  I'  Sr  has  replaced  I8r,  and  n'T  the  number  of  representa- 
tive paths  cutting  a  square  cm.  in  one  direction  has  replaced 
nr.  We  may  give  I'  Sr  any  value  we  please  above  a  limit 
determined  later,  and  determine  the  corresponding  value  of 
n'r  from  the  equation.  Since  rir  is  a  measure  of  the  chance 
of  a  molecule  crossing  a  plane  one  square  cm.  in  area  in 
moving  along  its  representative  path,  it  follows  from  the 
foregoing  equation  that  this  chance  is  inversely  proportional 
to  l'8r. 


.224    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

The  quantity  n'T  in  equation  (170)  may  be  expressed  in 
terms  of  other  quantities  which  are  of  interest.  Thus  if 
V'Sr  denote  the  average  velocity  a  molecule  would  have 
if  it  passed  over  its  representative  path,  instead  of  over  its 
actual  path,  it  can  be  shown  that 


_ 


along  the  same  lines  as  equation  (35)  was  obtained.     We 
also  have  directly  that 

7> 
TTt  v   fir 


where  t1 'ir  denotes  the  average  time  it  would  take  the  mole- 
cule to  pass  over  the  average  diffusion  path  l'&r,  if  it  passed 
over  its  representative  paths.  It  should  be  noted  that  V'8r 
is  not  equal  to  the  total  average  velocity  Vtr  of  the  molecule 
unless  I'sr  is  equal  to  lSr.  By  means  of  the  foregoing  two 
equations  equation  (170)  may  be  written  in  the  forms 

^-•^V* <m> 

and 


In  these  equations  we  may  give  l'dr  any  value  we  please 
above  a  limit  which  will  be  obtained  presently  and  determine 
from  them  the  corresponding  values  of  VfSr  and  t'Sr. 

Since  the  path  of  a  particle  is  undulatory,  and  the 
points  locating  the  diffusion  paths  lie  upon  the  actual  path 
of  the  particle,  it  is  evident  that  the  smaller  l'Sr  is  taken  the 
nearer  is  th«  -  representative  path  equal  to  the  actual  path,  and 
the  nearer  V'Sr  approaches  Vtr  the  total  average  velocity  of  the 
molecule.  Therefore  V'dr  cannot  be  smaller  than  Vtr,  and  l'Sr 
therefore  according  to  equation  (171)  not  smaller  than 


A  MOLECULAR  PATH  RELATION  225 

Wr/Vtr-  This  limiting  value  of  l'Sr  corresponds  to  the  value 
of  the  diffusion  path  lSr  defined  in  Section  41,  since  the 
representative  and  actual  molecular  paths  are  the  same  in 
length.  But  the  points  locating  the  path  lSr  cannot,  and  do 
not,  lie  on  the  actual  path.  It  follows  therefore  that  under 
the  foregoing  conditions  l'Sr  can  have  values  only  which  are 
somewhat  larger  than  the  value  of  lSr.  The  paths  are  evi- 
dently of  interest  only  under  these  conditions,  namely  in  that 
their  points  of  location  lie  on  the  actual  path  of  the  molecule. 

If  a  molecule  in  a  substance  moved  along  a  straight  line 
with  a  constant  velocity,  the  period  t'Sr  would  be  proportional 
to  l'Sr,  instead  of  proportional  to  (l'Sr)2  as  indicated  by 
equation  (172).  It  follows  therefore  that  a  molecule  pur- 
sues a  zigzag  course  in  a  substance.  It  is  interesting  to 
illustrate  equation  (172)  by  means  of  a  diagram  in  this 
connection.  Thus  let  abed  in  Fig.  18  denote  the  actual  path 
of  a  molecule  and  ac  and  ad  two  selected  mean  diffusion 
paths  of  which  ad  has  double  the  length  of  ac.  Now  it 
follows  from  equation  (172)  that  if  the  molecule  traveled 
over  its  representative  paths,  the  time  taken  in  passing 
over  the  path  ad  is  22  or  4  times  the  time  taken  in  passing 
over  the  path  ac.  And  since  the  molecule  takes  the  same 
time  in  passing  over  its  actual  path  as  over  its  representa- 
tive path  it  follows  from  the  figure  that  the  molecule  on 
the  average  takes  times  in  the  ratio  of  1  to  4  in  passing 
over  the  actual  paths  abc  and  abed,  where  ad  =  2ac. 

It  is  obvious  that  we  may  substitute  n'r,  n'e,  lfSr,  and 
l'6e  for  nr,  ne,  lSe,  and  lde  in  the  general  diffusion  equation 
(126),  where  the  former  symbols  have  meanings  of  the 
nature  just  considered.  The  equation  may  be  given  forms 
involving  V'Sr,  V'Se,  t'Sr,  and  t'Se  similarly  as  just  shown. 

The  points  on  the  actual  path  of  a  molecule  which  indi- 
cate the  location  of  the  diffusion  paths  may  be  selected  in 
a  different  way  than  the  foregoing,  which  is  simpler  and 
physically  more  definite.  Thus  we  may  select  the  points 


226     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

»  » 

so  that  each  line  joining  two  consecutive  points  is  equal  to 
the  same  length  instead  of  being  grouped  about  a  mean 
path  according  to  Clausius'  law,  and  that  any  direction  of 
a  line  in  space  is  equally  probable.  The  deduction  of  the 
diffusion  equation  is  then  simplified  since  Clausius'  prob- 
ability factor  need  not  be  introduced,  and  therefore  no 
integration  with  respect  to  z  is  necessary.  The  integral 
with  respect  to  6  (Section  42)  is  the  same  as  before,  and 


FIG.  18. 

introduces  the  factor  J  into  the  right-hand  side  of  the 
diffusion  equation,  which  otherwise  would  be  cancelled 
by  the  foregoing  integral.  The  coefficient  of  diffusion  in 
this  case  for  a  dilute  solution  is  therefore  given  by 

_  1  n"r  l"Sr 


Nr 


where  l"^  denotes  the  selected   diffusion  path,   and  n"r 
the  number  of  diffusion  paths  passing  through  an  area  of 


A  GENERAL  EXTENDED  DIFFUSION  EQUATION     227 

one  square  cm.  in  one  direction  per  second.  The  foregoing 
equation  may  be  written  in  the  forms 

V"    1" 
Dr  =  — *p* (174) 

and 

DT=l~t^       (175) 

similarly  as  before,  where  V"Sr  denotes  the  average  velocity 
the  molecule  would  have  if  it  passed  along  its  diffusion 
paths,  and  t"8r  the  average  time  it  would  take  to  pass  over 
a  single  diffusion  path.  It  can  be  shown  similarly  as  before 
that  l"8r  can  have  values  only  somewhat  greater  than  the 
average  diffusion  path  I8r. 

The  general  diffusion  equation  (corresponding  to  equa- 
tion (126))  may  be  written  in  this  case 

(l"&r)2Ne  dNr       (l"Se)2Nr  dNe 
c*n  j~  G*rr  j— 


where  the  meaning  of  the  different  symbols  is  evident  from 
what  has  gone  before. 

The  points  on  the  path  of  a  molecule  locating  the  dif- 
fusion paths  may  be  selected  in  a  third  way  which  is  of 
interest.  Thus  the  period  the  molecule  takes  to  pass  from 
one  point  to  the  next  may  be  taken  the  same.  Let  us  suppose 
in  this  connection  that  of  Nr  molecules  r  per  cubic  cm. 
nri  have  a  diffusion  path  l\  and  nr2  a  diffusion  path  /2,  and 
so  on,  corresponding  to  the  period  t'"sr.  We  may  consider 
each  of  these  sets  of  molecules  separately  and  apply  the 
preceding  result  to  obtain  expressions  for  the  diffusion. 
On  taking  into  account  that  the  concentration  gradient 

corresponding  to    the    molecules  nfl  is  ~  -^,  and  that 

J\  f  ax 

n    dN 
corresponding  to  the  molecules  nr%  is  -£-  -r^,  and  so  on,  it 


228     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

follows  at  once  that  the  coefficient  of  diffusion  of  a  dilute 
solution  is  given  by 


-  -  ._, 
'~6T'7,r         ~1VT        ~\-§~T^'   ' 

where  (l"fSr)2  denotes  the  mean  of  the  squares  of  the  dif- 
fusion paths.  The  corresponding  general  form  of  the  dif- 
fusion equation  may  now  be  written  down  without  any 
difficulty. 

It  will  readily  be  recognized  now  that  the  points  locating 
the  diffusion  paths  may  be  selected  in  other  ways,  in  fact 
according  to  any  law  we  please.  The  foregoing  three  ways 
are,  however,  the  only  ones  of  special  interest  and  importance. 

An  interesting  and  important  feature  of  the  foregoing 
equations  is  the  fact  that  either  the  period  or  the  diffusion 
path  may  be  given  any  value  we  please.  We  may  therefore 
immediately  calculate  the  diffusion  path  corresponding  to  a 
given  period,  or  vice  versa.  These  values  furnish  interesting 
information  about  molecular  motion.  A  set  of  such  cal- 
culations has  been  carried  out  in  the  next  Section  in  con- 
nection with  similar  equations  involving  viscosity. 

The  dependence  of  the  diffusion  path,  or  the  period, 
in  equation  (175)  on  more  fundamental  quantities  may  be 
obtained  by  equating  the  coefficient  of  diffusion  given  by 
the  equation  and  that  given  by  equation  (129),  and  substi- 
tuting for  I5r  from  equations  (130),  which  gives 


This  equation  may  also  be  written 


by  means  of  equation  (35).  Thus  for  a  selected  constant 
value  of  l"sr  the  period  varies  inversely  as  the  total  average 
velocity  Vtr  of  a  molecule  r,  as  we  might  expect..  The 


RELATIONS  OF  THE  PATH  PERIOD  229 

existence  of  the  apparent  volume  b'r  of  the  molecules  r 
and  e  has  the  effect  of  decreasing  the  period  from  what  it 
otherwise  would  be.  This  could  not  be  recognized  directly. 
The  existence  of  molecular  interference  has  the  same  effect 
since  &Sr  is  positive.  Also  an  increase  in  K'ST,  which  is 
proportional  to  the  diffusion  path  in  the  gaseous  state  at 
standard  pressure,  tends  to  decrease  the  period. 

The  period  may  be  expressed  in  terms  of  the  partial 
intrinsic  pressures  and  other  quantities  on  eliminating  nr 
from  equation  (178)  by  means  of  equation  (152). 

The  variation  of  l"Sr  for  a  given  selected  constant  period 
t"Sr  is  obtained  by  solving  the  foregoing  equations  with 
respect  to  I"  &. 

Equations  similar  to  the  foregoing  may  be  obtained 
corresponding  to  a  mixture  which  is  not  dilute.  Thus  on 
equating  the  expressions  for  5r  given  by  equations  (126) 
and  (176),  eliminating  ldr  and  I8e  by  means  of  equations 

(130),  and  equating  the  factors  of  -^  and  —=-?•  separately 

clx  dx 

to  zero,  which  holds  since  the  equation  is  an  identity,  we 
obtain  in  the  case  of  the  molecules  r  an  equation  which  is 
the  same  as  equation  (178)  having  the  factor 


introduced  into  the  right-hand  side.  Thus  we  see  that  in 
the  case  of  a  solution  of  molecules  r  in  e  which  is  not  dilute, 
an  increase  in  the  number  of  molecules  r  relative  to  the  mole- 
cules e  increases  the  period  t"8r.  The  existence  of  the  external 
molecular  volume  fte  of  the  molecules  e  decreases  the  period 
from  what  it  otherwise  would  be,  while  the  external  molecular 
volume  &T  of  the  molecules  r  increases  it. 

The  periods  corresponding  to  the  selected  paths,  or  vice 
versa,  may  approximately  be  calculated  from  equations 
(178)  and  (179)  on  determining  the  approximate  values 


,•230    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

of  the  quantities  on  their  right-hand  sides  by  means  of  the 
method  described  in  Section  42. 

54-  Extended  Forms  of  the  Viscosity  Equations. 

The  equations  for  the  viscosity  given  in  Sections  34  and 
35  may  be  given  extended  forms  similarly  as  the  diffusion 
equations  in  the  previous  Section.  Points  may  be  selected 
on  the  actual  path  of  each  molecule  and  these  joined,  and 
the  supposition  made  that  momentum  is  given  to  the  sub- 
stance parallel  to  its  motion,  or  abstracted  from  the  sub- 
stance, at  these  points  only  by  the  molecule.  The  line 
joining  two  consecutive  points  may  be  called  a  momentum 
transfer  distance  under  the  stated  conditions.  The  points 
of  location  of  the  transfer  distances  may  be  selected 
according  to  any  given  law;  there  are  three  ways  only, 
however,  which  are  of  interest,  and  which  were  used  in  con- 
nection with  diffusion.  Thus  we  may  suppose  that  the 
transfer  distances  are  grouped  about  their  mean  according 
to  Clausius'  law;  or  are  equal  to  each  other;  or  correspond 
to  the  same  period  for  the  molecule  to  pass  from  one  point 
to  the  consecutive  point,  the  points  in  each  case  being  so 
selected  that  any  direction  of  a  line  of  given  length  is  equally 
probable. 

In  the  first  case  the  form  of  the  viscosity  equation  re- 
mains the  same  as  that  of  equation  (83),  and  for  a  pure 
substance  r  may  be  written 


(180) 


where  /'  \r  denotes  the  mean  momentum  transfer  distance 
in  this  case,  and  n'r  the  number  of  representative  paths 
crossing  a  square  cm.  from  one  side  to  the  other  per  second. 
The  latter  quantity  is  given  by 


3    ' 


EXTENDED  GENERAL  VISCOSITY  EQUATION       231 

where  Vnr  denotes  the  velocity  the  molecule  would  have  if 
it  moved  along  its  representative  path,  this  quantity  being 

directly  given  by 

V 

v    •  --JE 

¥*ir—.f    > 
f  r,r 

where  t'^  denotes  the  period  of  the  average  transfer  distance. 
Equation  (180)  may  therefore  be  written  in  the  more  impor- 
tant form 


(181) 


The  equation  for  the  viscosity  of  a  mixture  of  molecules 
r  and  e  is  accordingly  given  by 


Nrmar  (I'^.a.     ,e 

---  ~~       •   •   '    (182) 


If  the  transfer  distances  are  taken  equal  to  each  other, 
it  can  be  shown  similarly  as  in  the  previous  Section  that 
the  corresponding  viscosity  equations  are  the  same  in  form 
as  equations  (181)  and  (182)  having  the  factor  \  introduced 
into  each  right-hand  side,  that  is,  we  may  write 

r,  =  ^y, (183) 

and 

_Nrmar  (l"r,r}2, 
rJ~ a =Fr r 


TV  '  TV 

where  l'\r  denotes  the  selected  constant  momentum  transfer 
distance  of  a  molecule  r,  t"^r  the  mean  of  its  periods,  while 
the  other  symbols  referring  to  the  molecules  e  have  similar 
meanings. 

If  the  average  period  for  a  pure  substance  in  equation 
(183)  is  taken  equal  to  the  period  in  the  case  of  equation 
(175)  applied  to  the  diffusion  of  a  molecule  in  molecules  of 


332     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

the  same  kind,  and  this  period  eliminated  from  the  two 
equations,  we  obtain  the  equation 


If  the  nature  of  the  motion  of  a  molecule  were  not  altered 
by  a  shearing  motion  given  to  the  substance,  l"v  would  be 
equal  to  l"Sr,  which  corresponds  to  the  normal  motion  of  a 
molecule.  The  foregoing  equation,  however,  indicates  that 
this  does  not  hold,  as  we  might  expect,  and  expresses  the 
effect  of  shearing  motion  on  the  value  of  l"v  for  unit 
velocity  gradient  in  the  substance. 

Since  the  viscosity  of  a  gas  is  independent  of  its  pressure, 
or  the  molecular  concentration,  it  follows  from  equation 
(183)  that  the  value  of  the  average  period  t"^  for  a  constant 
value  of  I"  ^  is  proportional  to  the  molecular  concentration 
NT.  This  shows  that  the  molecules  in  their  encounters 
deflect  each  other  from  their  paths,  and  therefore  the  actual 
path  of  a  molecule  between  two  points  is  greater  than  the 
straight  line  joining  them.  We  may  therefore  say  that  the 
period  a  molecule  takes  to  traverse  its  curved  and  zigzag 
path  between  two  points  a  constant  distance  apart  in  a 
gas  whose  pressure  is  varied,  is  proportional  to  its  chance  of 
encountering  another  molecule. 

The  viscosity  of  a  liquid  is  greater  than  that  correspond- 
ing to  the  gaseous  state,  and  hence  according  to  equation 
(183)  the  average  period  trrv  f°r  a  constant  value  of  V'v 
in  the  case  of  a  liquid,  is  smaller  than  proportional  to  Nr,  or 
smaller  than  the  chance  of  the  molecule  encountering  an- 
other molecule  along  its  path.  This  would  be  the  case  if 
the  average  velocity  of  a  molecule  in  the  liquid  state  is 
greater  than  in  the  gaseous,  which  would  decrease  the  time 
it  takes  the  molecule  to  pass  over  its  path.  This  fits  in  with 
results  obtained  previously.  .  . 


VALUES  OF  PERIODS  FOR  A  GIVEN  PATH 


233 


It  is  of  interest  to  obtain  by  means  of  equation  (183) 
for  different  substances  the  average  periods  t"nT  that  a  mole- 
cule takes  to  pass  from  one  point  to  another  one  mm.  apart, 
or  corresponding  to  /"„,=  .!  cm.  Table  XXVIII  gives  the 
values  of  the  periods  in  the  case  of  C02  at  different  pres- 


TABLE   XXVIII 


VALUES  OF  t"T  CORRESPONDING  TO  l"1)T  =  .l  CM. 

C02   AT   40°  C 

P  in 
atmos. 

Volume 
of  a  gram 
molecule. 

r,  10>. 

Nr  lQ2i. 

l"r,r  in 
seconds. 

70 

245.7 

200 

2.52 

1.49 

80 

172.2 

218 

3.61 

1.95 

85 

130.7 

269 

4.74 

2.08 

94 

85.35 

414 

7.26 

2.08 

100 

78.96 

483 

7.86 

1.92 

112 

73.24 

571 

8.48 

1.75 

ETHER 

t°  C. 

p- 

,,ia>. 

Arrio« 

*'V  in 

seconds. 

13.5 

.7214 

1779 

6.05 

.67 

99 

.6421 

1133 

5.39 

.94 

CHLOROFORM 

0 

1.5264 

3827 

7.93 

.66 

60 

1.4108 

2791 

7.33 

.84 

BENZENE 

15.4 

.8840 

4387 

7.38 

.36 

78.8 

.8145 

3000 

6.65 

.45 

234     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

sures  at  the  temperature  40°  C.,  and  of  three  liquids  at 
different  temperatures.  The  period  of  a  molecule  of  CC>2 
when  it  does  not  obey  the  gas  laws  evidently  does  not 
depend  much  on  the  molecular  concentration,  the  effect 
of  the  increase  of  concentration  being  probably  more  or 
less  balanced  by  the  increase  in  molecular  velocity.  The 
order  of  the  period  is  two  seconds.  In  the  case  of  liquids 
the  period  is  increased  with  an  increase  of  temperature, 
as  we  would  expect,  since  the  concentration  is  thereby 
decreased  which  decreases  the  molecular  velocity.  Its  value 
appears  to  decrease  with  an  increase  in  the  molecular  weight 
of  the  liquid,  and  is  of  the  order  of  one  second. 

In  the  case  of  a  gas  r  at  standard  temperature  and  pres- 
sure the  period  is  under  the  foregoing  conditions  given  by 
the  equation 

-77-     4.5X10-7mr 

t    r,r=~     ——     — ,         ....        (186) 

where  mT  denotes  the  relative  molecular  weight,  and  r?  the 
viscosity  which  in  this  case  is  independent  of  the  pressure. 
An  idea  of  the  dependence  of  the  period  on  the  nature  of 
the  gas  may  be  obtained  from  an  inspection  of  Table  X 
which  contains  values  of  TJ  and  m  for  a  number  of  different 
gases. 

If  the  period  is  kept  constant  it  can  be  shown  similarly 
as  in  the  previous  Section  that  the  viscosity  in  the  case  of 
a  pure  substance  r  is  given  by 

Nrmar  (I      gr) 


6          t' 


where  (Z"V)2  denotes  the  mean  of  the  squares  of  the  transfer 
distances  corresponding  to  the  constant  period  £'",„  and  an 
equation  consisting  of  similar  terms  may  be  obtained  for  a 
mixture  of  substances. 


THE  PATH  PERIOD  AND  OTHER  QUANTITIES     235 

1 

It  can  be  shown  in  the  same  way  as  in  the  previous  Sec- 
tion that  the  smallest  admissible  values  of  l\n  l'\n  and 
I"V  are  somewhat  larger  than  the  value  of  l^  defined  in 
Section  33. 

The  dependence  of  the  extended  transfer  distances  and 
periods  on  the  nature  rf  molecular  interaction  is  obtained 
on  equating  the  expression  for  the  viscosity  obtained  in 
this  Section  with  those  obtained  in  Section  34.  Thus  in 
the  case  that  the  transfer  distance  is  kept  a  constant  length 
we  obtain  from  equation  (183)  and  equations  (90),  (91), 
and  (92),  the  equations 


,,  «» 

' 

. 

C\Q(\\ 

' 


on  adding  the  suffix  r  to  the  symbols  of  the  latter  equations 
in  order  to  bring  the  notation  into  line  with  the  former 
equation,  and  taking  into  account  that  VrNrmar  =  mr,  where 

AT  =  5.087X10 

It  will  be  seen  from  these  equations  that  the  apparent 
molecular  volume  6r,  and  the  interference  function  $^r 
(which  is  positive),  decreases  the  period  from  what  it  other- 
wise would  be.  Thus  the  existence  of  molecular  forces  of 
repulsion,  which  prevents  the  molecules  approaching  each 
other  within  any  degree  of  closeness,  and  the  existence  of 
interference  of  the  molecules  of  the  substance  with  two 
interacting  molecules,  have  the  effect  of  decreasing  the 
period.  An  increase  in  the  total  average  velocity  Vtr  has 
the  effect  of  decreasing  the  period,  as  is  also  directly  evident. 


236    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

In  the  case  of  a  substance  not  obeying  the  gas  laws  an  in- 
crease in  velocity  at  constant  temperature,  it  may  be  noted, 
may  be  brought  about  by  an  increase  in  the  density  of  the 
substance  (Sections  17  and  29).  An  increase  in  K^,  which  de- 
creases with  an  increase  of  the  molecular  mass,  has  the  effect 
of  decreasing  the  period,  and  this  also  holds  for  the  volume 
VT  of  a  gram  molecule.  An  increase  in  the  molecular  forces 
of  attraction  would  give  rise  to  an  increase  in  the  intrinsic 
pressure  Pnr,  and  in  the  number  nr  of  molecules  crossing  a 
square  cm.  per  second,  and  thus  give  rise  to  a  decrease  of 
the  period.  Equation  (189)  of  the  three  equations  (188), 
(189),  and  (190),  indicates  best  the  dependence  of  the 
period  on  the  temperature  at  constant  volume.  It  decreases 
with  an  increase  of  temperature,  since  Vtr,  «,r,  and  ^ 
increase  with  an  increase  of  temperature  at  constant  volume 
(Sections  16,  17,  and  34),  while  br  is  probably  approximately 
independent  of  the  temperature. 

The  dependence  of  l'\T  on  the  fundamental  properties 
of  a  substance  for  a  constant  period  is  obtained  on  solving 
the  foregoing  equations  with  respect  to  lnv.  It  is  evident 
that  the  effect  of  the  changes  in  the  quantities  considered 
on  the  quantity  l'\r  is  opposite  in  direction,  and  less  (through 
the  extraction  of  square  roots  on  solving  the  equations  as 
indicated)  than  in  the  case  of  the  period. 

In  order  to  obtain  the  dependence  of  the  periods  on 
other  quantities  in  the  case  of  a  mixture  of  molecules  r 
and  e,  each  of  the  terms  on  the  right-hand  side  of  equation 
(184)  is  equated  with  one  of  the  two  terms  to  which  it  cor- 
responds on  the  right-hand  side  of  equation  (98),  and  the 
resulting  equation  transformed  by  means  of  equations  (99) 
and  (100).  Equations  similar  in  form  to  equations  (188), 
(189),  and  (190)  will  be  obtained  in  this  way. 

Approximate  values  of  the  periods  corresponding  to 
given  transfer  distances,  or  vice  versa,  in  the  case  of  a 
mixture,  may  be  obtained  from  these  equations  on  obtain- 


THE  PARTITION  OF  MOMENTUM   TRANSFERENCE      237 

ing  approximate  values  of  the  other  quantities  they  contain 
in  the  way  described  in  Section  35. 

In  connection  with  the  foregoing  investigation  the  fol- 
lowing remarks  may  help  to  clear  up  any  difficulties  encoun- 
tered. If  we  take  a  plane  parallel  to  the  motion  of  the 
substance  corresponding  to  the  velocity  Vi,  it  follows  from 
considerations  of  equilibrium  that  the  algebraical  momentum 
per  molecule  parallel  to  the  plane  on  the  average  is  equal  to 
Vima.  Therefore  on  considering  two  planes  corresponding 
to  the  velocities  V\  and  ¥2  of  the  substance,  it  follows  that 
the  molecules  crossing  each  plane  in  the  same  direction  will 
each  have  on  the  average  its  momentum  parallel  to  the 
planes  changed  from  V\ma  to  Vznia  on  passing  from  one 
plane  to  the  other.  This  will,  of  course,  not  hold  for  each 
molecule  considered  independently.  These  considerations 
show  that  we  may  suppose  that  momentum  is  given  by  a 
molecule  to  the  medium  and  abstracted  from  it  in  these 
planes  only,  which  may  be  taken  any  distance  apart.  This 
may  be  illustrated  by  considering  a  number  of  similar  par- 
allel planes  corresponding  to  the  velocities  FI,  Vi,  .  .  .  Ve  of 
the  substance.  The  momentum  transferred  from  the  first  to 
the  last  plane  by  a  molecule  is  equal  to  ( (Vi  —  F2)  +  (¥2  —  Fa) 
+  .  .  .  (Ve-i—Ve)}ma,  or  equal  to  (Fi— Fe)ma,  and  thus  the  in- 
termediate planes  have  no  effect  on  the  amount  of  momentum 
transferred  from  the  first  of  the  planes  to  the  other.  It 
follows,  therefore,  that  a  molecule  crossing  one  of  the  fore- 
going planes  may,  or  may  not,  be  taken  to  change  its  vis- 
cosity momentum  in  crossing  it,  just  as  we  please.  Also  a 
molecule  may  recross  a  number  of  times  each  of  two  consec- 
utive planes  between  the  two  points  lying  on  the  planes  at 
which  only  it  is  supposed  to  change  its  viscosity  momentum. 
The  foregoing  planes  may  therefore  be  taken  to  indicate 
the  location  of  the  various  points  defining  the  molecular 
paths  used  in  the  previous  investigation ;  and  hence  it  follows 
that  these  paths  are  permissible  to  use. 


238    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

It  should  be  pointed  out,  however,  that  the  representa- 
tive molecular  path  probably  cannot  assume  all  numerically 
possible  values  between  two  given  limits,  or  that  it  is  prob- 
ably a  recurring  discontinuous  function.  It  can  easily  be 
shown,  for  example,  that  the  values  of  the  path  I  for  a  part 
which  is  a  straight  line  do  not  fit  in  with  the  relation  l2/t  = 
constant.  But  the  total  region  of  discontinuity,  if  not  zero, 
is  probably  small  in  comparison  with  the  remaining  region. 

55.  Extended  Forms  of  the  Heat  Conduction 
Equations. 

Extended  heat  conduction  equations  may  be  obtained 
along  the  same  lines  as  the  extended  viscosity  and  diffusion 
equations  in  the  preceding  two  Sections.  We  may  suppose 
as  before  that  the  heat  transfer  distances  are  defined  by 
points  distributed  according  to  any  given  law  on  the  paths 
of  the  molecules.  The  transfer  distances  are  of  interest  only, 
however,  when  the  points  are  distributed  according  to  one 
of  the  three  ways  already  used. 

If  the  transfer  distances  are  grouped  about  a  mean 
according  to  Clausius'  law  the  heat  conductivity  equation 
for  a  pure  substance  r  may  immediately  be  written 


(191) 


according  to  Section  38,  where  V  CT  denotes  the  average 
heat  transfer  distance  which  may  have  any  value  above  a 
certain  limit  to  be  determined  presently,  n'T  denotes  the 
number  of  transfer  distances  crossing  a  square  cm.  in  one 
direction  per  second,  Sor  denotes  the  specific  heat  at  con- 
stant pressure  per  gram  and  mar  the  absolute  molecular 
weight.  If  Vcr  denotes  the  average  velocity  a  molecule  would 
have  if  it  passed  along  its  successive  transfer  distances,  and  ter 
the  time  it  would  take  to  pass  over  the  average  transfer 


THE  CONSTANT  PATH    CONDUCTION  EQUATIONS    239 

distance  (which  is  the  same  as  the  time  taken  to  pass  over 
the  actual  path),  we  have 


Hence  equation  (191)  may  be  written  in  the  form 


3  t'cr    > 

which  is  more  important.  The  heat  conduction  equation 
for  a  mixture  of  molecules  r  and  e  may  now  immediately 
be  written  down. 

If  the  distance  between  each  pair  of  consecutive  points 
is  taken  the  same  and  equal  to  l"Cr,  and  t"cr  is  the  average 
period  of  the  transfer  distance,  it  can  be  shown  along  the 
same  lines  of  reasoning  as  contained  in  the  preceding  two 
Sections  that 


for  a  pure  substance,  and 

_  NrmarSor  (l"cr)2  .  NemaeSoe  (I"*}*  . 

~<r  "FT    ~s~  ~W  ' 

for  a  mixture  of  molecules  r  and  e. 

On  the  other  hand  if  the  period  t'"T  is  kept  the  same  it 
can  be  shown  along  similar  lines  as  before  that 


~  j-argT  \l/      CT)  /lOP^ 

~6       PT'  ..... 

in  the  case  of  a  pure  substance  r,  where  (l"fcr)2  denotes  the 
mean  of  the  squares  of  the  corresponding  heat  transfer  dis- 


240    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

tances.  An  equation  consisting  of  the  sum  of  similar  terms 
holds  for  the  coefficient  of  conductivity  in  the  case  of  a 
mixture. 

It  can  be  shown  along  the  same  lines  as  in  the  preceding 
two  Sections  that  the  least  admissible  values  of  l'Cr,  l"cr, 
and  T77^.  are  somewhat  larger  than  the  value  of  lc  as  defined 
in  Section  38. 

If  the  period  in  equation  (193)  is  taken  equal  to  the 
period  in  equation  (183)  and  it  is  eliminated  from  the 
equations,  the  equation 


is  obtained.  This  equation  gives  some  information  about 
the  effect  of  a  shearing  motion  of  a  substance,  and  of  a  flow 
of  heat,  on  the  nature  of  the  motion  of  a  'molecule.  For 
the  quantity  lr\T  refers  to  the  motion  of  a  molecule  in  a 
substance  undergoing  a  shearing  motion  corresponding  to 
a  unit  velocity  gradient,  and  the  quantity  I"  CT  to  the  motion 
of  a  molecule  in  a  substance  through  which  a  flow  of  heat 
takes  place  corresponding  to  a  unit  heat  gradient.  If  the 
molecular  motion  were  not  affected  by  these  conditions 
l"cr  would  be  equal  to  l"^.  But  this  is  evidently  not  the 
case  according  to  the  foregoing  equation,  since  the  term 
under  the  radical  is  not  equal  to  unity.  According  to  Sec- 
tion 38  the  right-hand  side  of  the  equation  is  larger  than 
unity  when  the  substance  is  in  the  liquid  state,  and  smaller 
than  unity  when  in  the  gaseous  state.  Therefore  on  the 
average  the  line  joining  two  points  on  the  path  of  a  molecule 
in  a  liquid  for  a  constant  period  is  increased  when  a  velocity 
gradient  exists  in  the  substance  in  comparison  with  the 
magnitude  of  the  line  when  a  heat  gradient  exists,  and  the 
opposite  holds  in  the  case  of  a  gas. 

The  fact  that  the  internal  molecular  energy  of  a  complex 
molecule  is  several  times  greater  than  the  free  kinetic  energy 


A  GENERAL  FORM  OF  TRANSFER  FORMULAE     241 

of  translation  (Section  13),  gives  rise  to  a  transference  of 
energy  to  the  medium  by  the  molecule  in  moving  along  a 
heat  gradient,  or  vice  versa,  which  is  considerably  greater 
than  the  change  in  kinetic  energy.  This  will  have  the 
effect  of  changing  the  nature  of  the  path  of  the  molecule 
considerably  from  what  it  would  be  if  the  molecule  possessed 
no  energy  apart  from  its  kinetic  energy  of  translation. 
The  distortion  of  the  molecular  path  by  a  heat  gradient  is 
probably  in  the  main  caused  in  this  way. 

If  the  periods  are  taken  equal  in  equations  (193)  and 
(175)  and  eliminated  we  obtain  the  equation 


l"8r 

W*-\ 


(          . 


The  quantity  l"Sr  may  be  taken  to  correspond  to  the  diffusion 
of  a  molecule  in  a  substance  of  the  same  kind  instead  of  in 
a  mixture  in  which  molecules  r  possess  a  concentration 
gradient.  The  motion  of  a  molecule  in  the  former  case  is 
normal.  It  would  be  interesting  to  test  the  equation  under 
these  conditions. 

The  heat  transfer  distance  and  its  period  may  be  connected 
with  more  fundamental  quantities  in  the  same  way  as  similar 
quantities  in  the  preceding  two  Sections  in  the  case  of 
viscosity  and  diffusion. 

It  will  be  seen  that  the  diffusion,  viscosity,  and  heat 
conduction  expressions  given  in  this  and  the  preceding 
two  Sections  are  each  of  the  form  l2/t  multiplied  by  a  factor 
which  depends  on  the  conditions  under  which  the  molecules 
are  undergoing  motion  of  translation.  Each  of  the  quan- 
tities I  and  t  may  be  taken  to  a  certain  extent  as  arbitrary, 
but  not  at  the  same  time,  since  one  determines  the  other. 

A  set  of  equations  for  the  diffusion,  viscosity,  and  heat 
conduction  of  substances  will  now  be  developed,  which  are 
similar  to  the  equations  obtained  in  this  and  the  preceding 


242     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

* 

two  Sections,  and  which  involve  the  projection  of  the  motion 
of  a  molecule  along  a  line. 

56.   The    Constant    Period    Displacement    Dif- 
fusion Equation  and  its  Applications. 

Let  us  take  an  imaginary  plane  for  reference  at  right 
angles  to  the  diffusion  in  a  heterogeneous  mixture  and 
determine  the  number  of  molecules  of  one  kind  that  drift 
across  it  per  square  cm.  per  second.  The  motion  of  the 
molecules  will  be  expressed  in  terms  of  their  displacement 
at  right  angles  to  the  foregoing  plane  for  a  selected  period 
t.  It  will  be  evident  that  in  each  element  of  volume  half 
of  the  molecules  during  the  time  t  undergo  a  displacement 
in  one  direction,  while  the  other  half  undergoes  a  displace- 
ment in  the  opposite  direction.  Suppose  that  nrv  nT2J  nT3, 
.  .  .  ,  molecules  r  whose  displacements  are  respectively 
drt1  dT2)  dr3J  .  .  .  ,  during  the  period  tr  cross  the  plane  per 
square  cm.  in  the  direction  of  their  diffusion — say  from 
top  to  bottom.  The  molecules  of  displacement  dTl  come 
from  a  cylinder  of  height  dT}  and  unit  cross-section  stand- 
ing on  the  plane,  while  the  molecules  of  displacement  drz 
come  from  a  cylinder  of  height  dT2  standing  on  the  plane, 
etc.  Therefore  if  n'fl,  n'T2,  n'Tz .  .  .  ,  molecules  r  of  dis- 
placements dTl,  dT»  dr3,  .  .  .  ,  respectively  come  from  each 
cubic  cm.  of  the  mixture  we  have 

nri  =  n'ridri,  nTz=nfr2dr»  etc. 
We  also  have 

2«+rc'r2+  .  .  .)=Nr,       .     .     .     (198) 

since  each  molecule  undergoes  a  displacement  during  the 
period  tr,  one-half  of  the  molecules  undergoing  a  displace- 
ment in  one  direction  and  the  other  half  undergoing  a  dis- 
placement in  the  other  direction,  where  NT  denotes  the  con- 


MOLECULAR  DISPLACEMENT  ACROSS  A  PLANE    243 

centration  of  the  molecules  r.  The  total  number  of  molecules 
r  which  cross  the  plane  in  the  direction  of  diffusion,  or 
from  top  to  bottom,  during  the  time  tr  is  therefore  equal  to 


But  molecules  r  also  cross  the  plane  in  the  opposite 
direction.  Similarly  as  before  the  molecules  of  displacement 
dri  come  from  a  cylinder  of  height  dfl  and  unit  cross-section 
standing  on  the  plane,  and  the  molecules  of  displacement 
dTz  come  from  a  cylinder  of  height  dr2  standing  on  the  plane, 
etc.  We  may  suppose,  to  simplify  matters,  that  the  molecules 
r  of  displacement  dfl  which  cross  the  plane  of  reference  in 
the  direction  of  the  diffusion,  or  from  top  to  bottom,  come 
from  a  plane  at  a  distance  dri/2  from  the  plane  of  reference, 
while  the  molecules  of  displacement  dTl  crossing  in  the 
opposite  direction  come  from  a  plane  at  the  distance  drJ2 
on  the  opposite  side  of  the  plane  of  reference.  The  numbers 
of  these  two  sets  of  molecules  are  proportional  to  the  con- 
centrations of  the  molecules  at  the  starting  planes.  If  the 
concentration  at  the  upper  starting  plane  is  Nr,  that  at  the 

.     ,7       ,    dNr      i         dNT  .    Al_ 

lower  plane  is  NT—dTl  -r-  ,  where  -T—  is  the  concentration 
dx  dx 

gradient  of  the  molecules  r  measured  in  the  direction  of 
increase  of  concentration.     We  may  therefore  write  KNr 

and  K(  NT  —  dr.—  r^  )  for  the  numbers  of  these  two  sets  for 
\  dx  / 

migrating    molecules,    where    K    denotes    an   appropriate 
factor.    Since  nri  =  KNr  these  numbers  may  also  be  written 

nridri  dNr 
«,,  and  m—  ^  -fc, 

or 

,    ,      n'nd?n  dNr 
n'ridfl  and  n'ndn  --        -* 


,•244     MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

since  nTl  =  n'ridri.  Therefore  the  excess  of  molecules  r  of 
displacement  dTl  crossing  the  plane  of  reference  in  the  direc- 
tion of  diffusion  over  the  molecules  r  of  displacement  dri 
crossing  in  the  opposite  direction  in  the  time  tr,  is  equal  to 

n'nd?n  dNr 
Nr     dx' 

Similarly  it  can  be  shown  that  the  excess  of  molecules  r 
of  displacement  dT2  crossing  the  plane  is  equal  to 


Nr     dx' 

and  so  on.  Therefore  the  total  excess  of  molecules  r  that 
cross  the  plane  in  one  second  due  to  the  displacements  they 
undergo,  is  equal  to 

•  (2n'nd?n+2n'r#n+  .  .  .  )  ldNr  =  d?rdNr 
2(2n'T1+2n'r2+  .  .  .  )      tr  dx      2tr  dx  ' 

by  the  help  of  equation  (198),  where 

(2n'nd2n+2n'r2d2n+  .  .  .)_^- 
(2n'n+2n'n+  .  .  .  ) 

the  mean  of  the  squares  of  the  displacements.  Similarly 
it  can  be  shown  that  the  excess  of  molecules  e  that  cross 
the  plane  in  the  opposite  direction  per  second  due  to  the 
displacements  they  undergo  is  equal  to 

QdN, 

2te  dx  ' 


where  d2e  denotes  the  mean  of  the  squares  of  the  displace- 

dN 
ments  corresponding  to  the  period  te,  and  -—^  the  con- 

CtOu 

centration  gradient  of  the  molecules  e  measured  in  the 


DISPLACEMENT  DIFFUSION  EQUATION  245 

direction  of  increase  of  concentration,  and  thus  in  the 
opposite  direction  that  the  -gradient  of  the  molecules  r  is 
measured.  Thus  on  the  whole  there  is  a  gain  of  molecules 
r  and  e  per  second  below  the  plane  equal  to 


2tr  dx      2te  dx' 

But  the  space  occupied  by  these  molecules  is  not  zero. 
A  portion  of  the  mixture  must  therefore  be  transported 
bodily  across  the  plane  to  make  room  for  the  foregoing 
molecules.  This  may  be  taken  into  account  in  exactly  the 
same  way  as  in  Section  42.  It  will  appear  from  that  investi- 
gation that  the  total  migration  5r  of  molecules  r  across  the 
plane  per  second  is  accordingly  given  by 


2tr     dx        2te     dx 

where  $r  and  de  denote  the  external  molecular  volumes  of 
gram  molecules  of  molecules  r  and  e  in  the  mixture.  The 
diffusion  of  the  molecules  e  is  obtained  by  interchanging 
the  suffixes  r  and  e  in  the  foregoing  equation. 

In  the  case  of  a  dilute  solution  of  molecules  r  in  e  the 
foregoing  equation  becomes 

d?rdNr 


The  coefficient  of  diffusion  Dr  in  this  case  is  therefore  given 
by  _ 

A=J (201) 

Thus  if  the  molecular  displacements  in  a  mixture  for  a  given 
period  could  be  measured  the  diffusion  could  be  calculated. 


^246    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

This  is  of  course  impossible  in  the  case  of  molecules,  but 
can  be  carried  out  in  the  case  of  Brownian  particles.  Exam- 
ples of  such  investigations  will  be  considered  further  on. 

The  foregoing  equations  are  limited  according  to  the 
following  considerations:  Suppose  that  the  displacements 
of  the  molecules  of  a  cubic  cm.  of  matter  were  observed  for 
several  consecutive  periods  each  equal  to  tr.  The  value  of 
d2T  for  each  set  of  observations  should  be  the  same,  since  the 
equations  are  independent  of  the  time  of  beginning  of  a 
set  of  observations.  Now  this  is  the  less  likely  to  be  realized 
the  smaller  the  value  of  tr.  Hence  there  is  a  limiting  value 
of  tr  for  lower  values  of  which  the  equations  do  not  hold. 
This  limiting  value  can  definitely  be  determined  as  will 
now  be  shown.  If  the  path  of  each  molecule  consisted  of  a 
number  of  straight  lines  joined  at  various  angles,  and  their 
lengths  were  grouped  about  a  mean  length  according  to 
Clausius'  probability  law  given  in  Section  31,  while  each 
inclination  of  a  line  in  space  were  equally  probable,  each 
line  or  molecular  free  path  would  be  independent  of  the 
preceding  path.  The  corresponding  displacements,  or  pro- 
jections of  the  free  paths  on  to  an  axis,  would  therefore  also 
be  independent  of  each  other.  But  this  would  evidently  not 
hold  for  displacements  corresponding  to  a  fraction  of  a  free 
path.  Now  the  average  of  the  displacements  dr,  which  is 
given  by 


and  the  average  free  path  ldr  are  connected  by  the  equation 

4d~r  =  kr (202) 

according  to  Section  44.  Hence  it  follows  that  a  period 
tr  is  inadmissible  if  the  corresponding  average  displacement 
has  a  smaller  value  than  given  by  the  foregoing  equation. 


THE  INFERIOR  LIMIT  OF  THE  DISPLACEMENT    247 

This  would  be  indicated  in  practice  by  the  values  of  tr  and 
d?r  not  fitting  in  with  equations  (200)  and  (199).  But  in 
practice  the  molecules  do  not  move  along  a  series  of  lines 
joined  together,  especially  in  a  liquid.  We  may,  however, 
(Sections  41  and  42)  find  a  series  of  rectilinear  paths  lying 
close  to  the  actual  path  equal  to  it  in  length,  which  indicate 
the  average  directions  of  different  portions  of  the  actual 
path  as  shown  in  Fig.  14,  and  whose  lengths  are  grouped 
about  a  mean  length  according  to  Clausius'  law,  and  whose 
directions  in  space  are  unrestricted.  The  parallel  planes 
passing  through  the  points  of  connection  of  these  paths 
divide  the  actual  path  into  a  number  of  parts  which  we  may 
take  as  being  independent  of  each  other  as  the  correspond- 
ing rectilinear  paths.  The  smallest  period  tr  admissible 
in  equations  (199)  and  (200)  would  then  be  that  correspond- 
ing to  the  average  displacement  which  satisfies  equation 
(202),  where  the  symbol  I8r  now  denotes  the  average  dif- 
fusion path  as  defined  and  used  in  Sections  41  and  42. 
Smaller  values  of  tr  and  the  corresponding  values  of  d?r 
will  therefore  not  satisfy  equations  (199)  and  (200). 

This  result  may  be  obtained  in  a  somewhat  different 
way.  Let  us  suppose  as  before  that  a  molecule  moves  along 
a  series  of  rectilinear  paths.  Consider  the  displacement 
corresponding  to  a  portion  of  a  rectilinear  path  for  a  period 
£,  which  is  much  smaller  than  the  period  of  the  path.  The 
corresponding  displacement  d\  is  then  proportional  to  t\, 
or  equal  to  tf!lt  where  C,  is  a  constant.  .  The  value  of 
d2r/tr  in  equation  (200)  under  these  conditions  is  therefore 
equal  to  tTC2,  where  €2  is  a  constant.  But  this  is  obviously 
impossible,  and  the  period  tr  can  therefore  be  given  values 
only  equal  to  or  greater  than  the  period  of  the  average  path. 
Since  the  actual  path  of  a  molecule  may  be  on  the  average 
represented  by  such  a  series  of  rectilinear  paths,  the  same 
result  holds  in  reference  to  the  period  of  the  average  rep- 
resentative path.  The  deviation  of  the  ratio  from  satis- 


248    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

•    .. 

fying    equation  (200)  on  decreasing  tr  would,  however,  be 
more  gradual  in  this  case. 

If  t'r  denote  the  smallest  admissible  period  that  satisfies 
equation  (200)  and  d'r  the  corresponding  average  of  the 
displacements,  the  total  average  velocity  Vtr  of  a  molecule 
r  is  given  by 


(203) 


by  the  help  of  equation  (202),  since  t'T  is  then  the  period 
of  the  average  diffusion  path  ldr,  and  the  representative 
and  actual  path  of  a  molecule  are  equal  in  length  (Section 
41).  This  equation  may  be  used  to  find  the  velocity  of  a 
colloidal  particle,  as  will  be  described  in  Section  60.  The 
number  of  times  nr  per  second  that  a  square  cm.  is  crossed 
by  molecules  r  in  one  direction  is  given  by 

_NrVtr_4Nrdrr 

Wr—  -3-      -gjT-,  -      •      •       (204) 

according  to  equations  (203)  and  (35). 

The  foregoing  remarks,  and  equations  (203)  and  (204), 
apply  also  to  a  mixture  of  molecules  r  in  e  which  is  not  a 
dilute  solution. 

The  largeness  of  the  value  that  may  be  taken  for  the 
period  tr  is  limited  somewhat  by  the  fact  that  in  the  deduc- 
tion of  equation  (199)  it  is  assumed  that  the  number  of 
molecules  undergoing  a  displacement  d  at  each  of  two 
equal  elements  of  volume  in  a  heterogeneous  mixture  at  a 
distance  d  apart  is  proportional  to  the  concentration  of  the 
molecules.  This  will  of  course  not  hold  when  the  difference 
between  the  concentrations  in  the  two  volumes  is  large. 
This  difference  evidently  depends  on  the  concentration 
gradient  of  the  mixture.  The  admissible  values  of  the 
displacement  and  period  therefore  depend  on  the  concen- 


THE   MOBILITY  IN   TERMS  OF  DISPLACEMENTS    249 

tration  gradient,  and  increase  with  decrease  of  the  gradient. 
The  same  remarks  apply  to  the  limits  of  l'Sr  in  Section  53. 

On  equating  the  coefficients  of  diffusion  given  by  equa- 
tions (201)  and  (175),  and  eliminating  (ln  br)2/t"  &T  from  the 
resultant  equation  by  means  of  equation  (178),  or  (179), 
we  obtain  equations  similar  in  form  to  the  latter  equations, 
which  express  d2r/tr  in  terms  of  fundamental  quantities. 
The  same  remarks  apply  to  these  equations  as  were  made 
in  connection  with  equations  (178)  and  (179). 

Equation  (201),  which  applies  to  a  dilute  solution,  has 
previously  been  deduced  *  by  Einstein  by  a  different  method, 
without  obtaining,  however,  its  exact  limitations.  The 
method  that  I  have  developed  is  more  general  and  funda- 
mental since  it  gives  diffusion  formulae  for  any  relative 
concentration  of  the  constituents  of  the  mixture  without 
referring  to  its  state,  and  gives  the  exact  limitations  of  the 
formulae.  It  should  be  noted  that  no  restrictions  whatever 
are  made  in  the  investigation  in  connection  with  the  molec- 
ular velocities.  By  means  of  this  method  other  diffusion 
formulae,  and  formulae  for  the  viscosity  and  conduction  of 
heat,  can  be  obtained  in  terms  of  molecular  displacements. 
These  are  given  in  subsequent  Sections. 

An  application  of  the  foregoing  results  to  the  Brownian 
motion  of  colloidal  particles  will  now  be  given. 

(a)  On  equating  the  values  of  the  coefficients  of  dif- 
fusion given  by  equations  (201)  and  (147)  the  equation 


(205) 


is  obtained,  which  expresses  the  coefficient  of  mobility 
of  a  particle,  or  the  velocity  under  the  action  of  unit  force, 
in  terms  of  other  quantities.  If  the  particle  is  spherical  in 
shape  and  its  size  is  so  large  that  its  mobility  obeys  Stokes7 

*  Ann.  der  Phys.,  19,  pp.  371-381  (1906). 


.250    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

law,    the    coefficient   of   mobility   given   by    the   foregoing 
equatiori  and  equation  (166)  may  be  equated  giving 


N  ,— (206) 


where  r  denotes  the  radius  of  the  particle,  and  77  the  vis- 
cosity of  the  medium.  This  equation  has  been  applied  to 
the  Brownian  motion  of  colloidal  particles  of  different 
substances  in  different  solvents,  the  values  of  _dr  for  a  selected 
value  of  tr  being  directly  observed  by  means  of  the  ultra- 
microscope  as  described  in  Section  44,  or  recorded  photo- 
graphically by  means  of  an  appropriate  arrangement. 
Some  of  the  earlier  observations  did  not  fit  in  well  with  the 
equation.  This  was  probably  due  to  some  extent  that  the 
period  was  taken  below  the  least  admissible  value,  the 
limitations  of  the  equation  not  being  known.  Later  experi- 
ments happen  to  have  been  carried  out  for  larger  periods 
and  consequently  gave  a  better  agreement.  It  is  also 
hardly  likely  that  the  colloidal  particles  obtained  by  spark- 
ing between  various  metal  electrodes  placed  in  a  liquid 
should  in  all  cases  be  exactly,  or  approximately,  spherical 
in  shape  as  required  by  the  equation.  We  would  expect 
that  the  geometrical  shape  of  the  particles  should  depend 
a  good  deal  on  the  conditions  of  the  sparking.  A  good 
confirmation  of  Einstein's  equation  was  recently  obtained 
by  Nordlund,*  who  used  particles  obtained  by  sparking 
mercury  in  a  liquid.  These  particles  are  bound  to  be  spherical 
in  shape  since  they  would  be  in  the  liquid  state,  and  hence 
Stokes'  law  would  hold.  Nordlund  determined  N,  the 
number  of  molecules  in  a  gram  molecule,  by  means  of  the 
equation,  and  obtained  the  value 

5.91  X1023, 
*  Zs.  f.  Phy.  Chem.,  87,  pp.  40-62  (1914). 


RATIO  SQUARE  OF  DISPLACEMENT  TO  PERIOD     251 


which  is  in  close  agreement  with  the  values  obtained  by 
other  methods. 

Einstein's  formula  has  been  exhaustedly  tested  by 
Millikan*  in  the  case  of  particles  suspended  in  a  gas,  and 
found  to  be  in  complete  agreement  with  the  facts. 

According  to  equation  (201)  the  value  of  —  is  constant 

tr 

for  different  values  of  dr  and  tr.    This  is  necessary  to  test 
before   using   equation    (206).       Table   XXIX   shows    the 

TABLE  XXIX 


Periods. 

Z  d2T. 

4 

u 

1.039 

1.039 

2t 

2.458 

1.229 

3t 

3.957 

1.319 

4t 

5.264 

1.416 

5t 

6.685 

1.337 

61 

7.963 

1.327 

Wt 

13.528 

1.353 

agreement  obtained  by  Nordlund  in  the  investigations  just 
described.    It  will  be  seen  that  for  periods  of  the  displace- 


ment dr  from  3£  to  Wt  inclusive  Z— -  is  practically  constant 

tr 

where  Z  is  a  constant.  The  values  for  the  periods  It  and  2t 
do  not  fit  in  well  with  the  others,  and  they  are  therefore 
probably  inadmissible  values  along  the  lines  previously 
explained.  (See  also  Section  60.)  The  result  is  of  interest 
and  importance  since  it  is  evidence  of  the  soundness  of  the 
deduction  of  equation  (199),  and  incidentally  of  the  deduc- 
tion of  the  equations  in  Sections  57  and  58,  and  of  the 

*  The  Electron,  by  R.  A.  Millikan,  the  University  of  Chicago  Science 

Series. 


252    MISCELLANEOUS  APPLICATIONS,   CONNECTIONS     - 

equations  in  Sections  53,  54,  and  55,  each  of  which  involves 
an  indeterminate  linear  magnitude  connected  with  the 
actual  path  of  a  molecule,  and  its  corresponding  indeter- 
minate period. 

57.  The  Constant  Displacement  Diffusion  Equa- 
tion. 

Instead  of  considering  the  displacement  of  the  mole- 
cules for  a  selected  period  /,  we  may  consider  the  different 
periods  of  the  molecules  corresponding  to  a  selected  displace- 
ment d.  This  case  may  be  treated  along  similar  lines  as 
that  in  the  previous  Section.  Suppose  that  n'Tl  molecules 
r  per  cubic  cm.  have  a  period  tri  in  undergoing  a  displace- 
ment dr  in  one  direction  at  right  angles  to  the  plane  of 
reference,  and  that  n'T2  molecules  have  a  period  tra  etc. 
Each  of  these  sets  of  molecules  may  be  treated  independ- 
ently along  the  same  lines  as  the  investigation  in  the  pre- 
ceding Section.  It  follows  immediately  then  from  the 
previous  Section  that  the  gain  per  second  in  molecules  r 
having  the  period  tri  below  the  plane  of  reference  is  equal 
to 

n'nd?rdNr 

dx> 


and  the  gain  in  molecules  r  having  a  period  tT2  is  equal  to 


and  so  on.    The  total  gain  in  molecules  r  is  therefore 

(i  

~£+~£+  '  *  '  W'S"*    2  r~S?> 

where 

~~\        r   /     T"  •   •   •    [TF; 
Ti  ^rj  I  -iVf 


A  DISPLACEMENT  DIFFUSION  EQUATION         253 

the  mean  of  the  reciprocals  of  the  periods.  Similarly  it 
can  be  shown  that  the  gain  in  molecules  e  below  the  plane 
of  reference  is 


2      dx' 

where  de  is  a  selected  displacement  and  te~  1  the  mean  of 
the  reciprocals  of  the  corresponding  periods.  The  gain  in 
molecules  r  and  e  due  to  their  displacements  is  therefore 


~ld2r  dNr       te~ 


2      dx         2       dx' 

But  the  volume  of  these  molecules  is  not  zero,  and  if  this  is 
taken  into  account  similarly  as  in  the  previous  Section  we 
obtain  that  the  diffusion  dr  of  the  molecules  r  is  given  by 


*  _ 

8r~ 


2  NN  re  dx       e        er  dx 


where  &e  and  #r  denote  the  external  molecular  volumes  of 
gram  molecules  of  molecules  e  and  r  respectively  in  the 
mixture. 

In  the  case  of  a  dilute  solution  of  molecules  r  in  mole- 
cules e  the  foregoing  equation  becomes 


fr      T 

and  hence  the  coefficient  of  diffusion  is  given  by 

Dr  =  '^f (209) 

It  can  be  shown  along  the  same  lines  as  in  the  previous 
Section  that  the  smallest  admissible  value  of  dr  in  equation 
(209)  corresponds  to  its  value  in  the  equation 

4dr=l5r,  (210) 


254    MICSELLANEOUS  APPLICATIONS,  CONNECTIONS 

where  I8r  denotes  the  average  diffusion  path  according  to 
Section  41.  If  d'  T  denotes  the  smallest  admissible  value  of 
dr  and  t'T  the  average  of  the  corresponding  periods,  the 
total  average  velocity  Vtr  of  a  molecule  r  is  given  by 

F,r=^=4|r,     .....     (211) 

tr  t  T 

by  means  of  the  foregoing  equation,  since  the  average  dif- 
fusion path  is  equal  to  the  corresponding  actual  path.  The 
number  of  times  nr  a  square  cm.  is  crossed  per  second  by 
molecules  r  in  one  direction  is  given  by 


according  to  equations  (211)  and  (35).  The  foregoing 
remarks  and  equations  also  hold  when  the  mixture  is  not  a 
dilute  solution  of  molecules  r  in  e. 

On  equating  the  expressions  for  Dr  given  by  equations 
(209)  and  (147)  we  obtain  the  equation 


(213) 


which  may  be  used  to  determine  the  coefficient  of  mobility 
Mr  of  a  colloidal  particle.  This  equation  and  equation 
(166)  give  the  equation 


RT    1 

.....     (214) 


which  depends  on  Stokes'  law,  and  may  be  used  in  the 
same  way  as  equation  (206). 

It  will  now  be  recognized  that  an  expression  for  the 
diffusion  may  be  obtained  along  the  same  lines  corresponding 
to  any  law  of  selection  of  the  displacements,  or  of  the  periods. 


DISPLACEMENT  OF   MOMENTUM  255 

58.  The  Constant  Period  and  Constant  Displace- 
ment Viscosity  Equations. 

The  viscosity  of  a  substance  also  may  be  expressed  in 
terms  of  molecular  displacements  at  right  angles  to  a  plane. 
Let  us  first  obtain  a  formula  for  the  viscosity  in  terms  of 
displacements  corresponding  to  a  constant  period. 

(a)  Let  the  plane  of  reference  be  taken  parallel  to  the 
motion  of  the  substance,  and  let  us  suppose  that  the  velocity 
gradient  is  unity,  and  for  convenience  of  reference  from  top 
to  bottom.  We  will  adopt  the  same  notation  as  in  Section 
56,  and  suppose  that  we  are  dealing  with  a  pure  substance 
of  molecules  r  instead  of  with  a  solution  of  molecules  r 
in  molecules  e.  Let  us  suppose  that  a  molecule  at  the  begin- 
ning of  a  displacement  (which  takes  place  at  right  angles 
to  the  motion  of  the  medium)  abstracts  the  momentum 
V\ma  from  the  medium,  where  V\  denotes  the  velocity  of 
the  medium  in  a  plane  parallel  to  its  motion  passing  through 
the  molecule  whose  absolute  mass  is  mar.  This  momentum, 
we  will  suppose,  is  transferred  to  the  medium  at  the  end 
of  the  displacement.  Now  it  follows  from  considerations  of 
equilibrium  that  the  displacement  of  a  molecule  in  one 
direction  from  one  plane  to  another,  both  of  which  are 
parallel  to  the  motion  of  the  medium,  is  accompanied  by  the 
displacement  of  another  molecule  from  one  plane  to  the 
other  in  the  opposite  direction.  Therefore  if  V\  and  ¥2 
denote  the  velocities  of  the  medium  at  two  planes  taken  a 
distance  d  from  each  other,  the  momentum  transferred 
from  one  plane  to  the  other  by  a  molecule  and  its  associate 
molecule  is  (V\  —  V2)ma,  or  dma,  since  the  velocity  gradient 
of  the  medium  is  unity,  or  (V\  —  Vz)/d=\.  Since  nri  mole- 
cules of  displacement  dri  cross  the  plane  of  reference  per 
square  cm.  in  each  direction  during  the  period  tr,  or  n'Tldri 
molecules,  where  n'Tl  denotes  the  number  of  molecules  in 
a  cubic  cm.  undergoing  a  displacement  drt  in  one  direction, 


256    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

it  follows  that  these  molecules  transfer  the  momentum 
nfrid2riniar  across  the  plane.  Similarly  it  follows  that  the 
molecules  undergoing  a  displacement  dri  transfer  the  mo- 
mentum n'rf-r^nar  during  the  period  tr  across  the  plane 
per  square  cm.  and  so  on.  The  total  momentum  trans- 
ferred per  second  per  square  cm.  of  the  plane,  which  is  the 
coefficient  of  viscosity  r?,  is  therefore  given  by 

_  Nrmar  1 2n'nd2n + 2rir42r*+  .  .  .  " 


or 

'-^r* (215) 

where  d2r  denotes  the  mean  of  the  squares  of  the  displace- 
ments of  the  molecules  of  a  cubic  cm.  during  the  period  tr. 

It  will  not  be  difficult  to  see  that  in  the  case  of  a  mixture 
of  molecules  r  and  e  the  coefficient  of  viscosity  is  given  by 
the  equation 


_        rar  ,  Ned2emae  /0  !  £>. 

~       ~~~~ 


where  d2e  denotes  the  mean  of  the  squares  of  the  displace- 
ments of  the  molecules  e  in  a  cubic  cm.  during  the  period  te. 
It  can  be  shown  in  a  similar  way  as  in  Section  56  in  con- 
nection with  diffusion  that  the  smallest  admissible  value 
of  d2r  in  equations  (215)  and  (216)  corresponds  to  the 
value  of  the  mean  of  the  displacements  dr  which  satisfies 
the  equation. 

43",  =  ^,    .......     (217) 

where  l^  denotes  the  mean  momentum  transfer  distance 
of  a  molecule  r  as  defined  in  Section  33. 


COMPARISON  OF  DIFFERENT   DISPLACEMENTS    257 

If  d'T  denotes  the  smallest  admissible  mean  of  the  dis- 
placements and  t'  r  the  corresponding  period,  the  total  aver- 
age velocity  is  given  by 


(218) 


while  the  number  of  molecules  nr  crossing  a  square  cm. 
in  one  direction  is  given  by 


obtained  along  the  same  lines  as  similar  equations  in  Sec- 
tion 56. 

If  equation  (201)  is  supposed  to  apply  to  the  diffusion 
of  a  molecule  in  a  substance  of  the  same  kind  and  the  period 
is  taken  the  same  as  the  period  in  equation  (215)  and 
eliminated  from  the  equations  we  obtain  the  equation 

=  =T^v7-  (220) 

d2Sr     DTNrmar' 

where  the  suffixes  8  and  77  have  been  added  to  the  displace- 
ments to  indicate  what  each  refers  to.  According  to  this 
equation  the  ratio  on  the  left-hand  side  is  not  in  general 
equal  to  unity.  Thus  the  displacement  of  a  molecule  along 
an  axis  is  affected  by  a  shearing  motion  of  the  substance  at 
right  angles  to  the  axis. 

In  the  case  of  a  dilute  solution  of  molecules  e  in  r  the 
value  of  d?r  will  very  approximately  be  the  same  as  for  the 
solvent  in  the  pure  state,  and  may  accordingly  be  obtained 
for  any  selected  period  tr  from  equation  (215).  The  value 
of  d?~e  may  then  be  obtained  for  any  selected  period  te  from 
equation  (216).  It  would  furnish  interesting  information 
to.  compare  this  value  of  d2e  with  that  of  a  pure  substance 


258    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

of  molecules  e,  or  if  the  solute  consists  of  colloidal  particles, 
to  compare  the  value  with  that  observed. 

If  rj  is  eliminated  from  equation  (215)  by  means  of  one 
of  the  equations  (90),  (91),  and  (92),  an  equation  is  obtained 
which  expresses  the  ratio  d27/tr  in  terms  of  more  fundamental 
quantities,  the  equation  being  similar  to  one  of  the  equa- 
tions (188),  (189),  and  (190). 

If  we  write  te  =  tr  and  d2e  —  d2T  in  equation  (216)  it  gives 
for  any  selected  value  of  tr  a  value  of  d2r  which  in  a  sense  is 
a  mean  value  of  d2e  and  d2T.  It  would  be  interesting  to  com- 
pare this  value  with  that  of  the  substance  r  in  the  pure  state. 

The  values  of  d2r  and  d2e  in  equation  (216)  may  be 
determined  for  any  selected  values  of  tr  and  te  by  equating 
each  of  the  two  terms  on  the  right-hand  side  of  the  equa- 
tion with  the  corresponding  term  on  the  right-hand  side 
of  equation  (98),  and  calculating  the  other  quantities  in- 
volved in  the  way  described  in  Section  35. 

It  can  easily  be  proved  along  the  same  lines  as  the 
preceding  investigation  that  if  the  periods  of  the  molecules 
in  a  cubic  cm.  for  a  selected  displacement  dr  are  observed, 
the  coefficient  of  viscosity  is  given  by 

77  =    Vrd2rt^marj (221) 

where  tr~l  denotes  the  mean  of  the  reciprocals  of  the  periods 
corresponding  to  the  displacement  dr  and  is  given  by 

n     2nf_n 

'          ' 


(2rin+2n'n+  .  .  .  ) 
In  the  case  of  a  mixture  of  molecules  r  and  e  we  have 

The  smallest  admissible  value  of  dr  in  equation  (221) 
satisfies  the  equation 

UT  =  lv,    .    .       ,    .    ,    ,     (223) 


DISPLACEMENT  OF  HEAT  ENERGY  259 

which  follows  similarly  as  before.  The  formulae  for  the 
total  average  velocity  and  the  number  of  molecules  r  crossing 
per  second  a  square  cm.  from  one  side  to  the  other  in  terms 
of  the  smallest  admissible  displacement  d'r  and  the  cor- 
responding mean  of  the  periods  t'r  are  the  same  in  form  as 
equations  (218)  and  (219).  Similar  relations  apply  in  the 
case  of  the  molecules  in  a  mixture  of  molecules  r  and  e. 

59.  The    Constant   Period    and    Constant   Dis- 
placement Heat  Conduction  Equations. 

We  will  now  find  an  expression  for  the  coefficient  of 
conduction  of  heat  in  terms  of  molecular  displacements, 
using  the  same  notation  as  before,  and  taking  the  plane  of 
reference  at  right  angles  to  the  flow  of  heat,  which  we  will 
suppose  takes  place  from  top  to  bottom.  A  formula  involv- 
ing the  displacements  for  a  selected  constant  period  will 
be  first  obtained.  Let  us  suppose  that  a  molecule  r  at  the 
beginning  of  a  displacement,  (which  takes  place  parallel 
to  the  flow  of  heat  at  right  angles  to  the  plane  of  reference) 
abstracts  the  energy  TiSmr  from  the  medium  and  transfers 
it  to  the  medium  at  the  end  of  the  displacement,  where 
TI  denotes  the  absolute  temperature  of  the  medium  at  the 
beginning  of  the  displacement,  and  Smr  the  corresponding 
internal  specific  heat  at  constant  pressure  per  molecule. 
Since  no  matter  is  transferred  from  one  plane  to  another, 
each  of  which  is  at  right  angles  to  the  flow  of  heat,  it  fol- 
lows that  the  displacement  of  the  foregoing  molecule  is 
accompanied  by  the  displacement  of  another  molecule  in 
the  opposite  direction,  both  displacements  lying  between 
the  same  two  parallel  planes.  Therefore  if  TI  and  TI  are 
the  absolute  temperatures  in  descending  order  of  magni- 
tude of  the  medium  in  these  planes  the  energy  T\Smr—  T2Smr 
is  transferred  to  the  lower  plane  by  a  pair  of  correspond- 
ing molecules  during  the  period  tr.  If  the  distance  between 


-•260    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

the  planes  is  d  and  the  heat  gradient  is  equal  to  unity 
Ti  —  T2/d=l,  and  the  amount  of  heat  transferred  may  be 
written  dSmr.  Since  nri  molecules  of  displacement  dri  cross 
the  plane  of  reference  in  both  directions  during  the  period 
tr,  which  number  (Section  56)  is  equal  to  n'ridri,  it  follows 
that  these  molecules  transfer  the  energy  nfrid2TlS^T  per 
square  cm.  across  the  plane.  Similarly  it  follows  that 
the  molecules  of  displacement  dzr  transfer  the  energy 
n'rtd2rj$mr  per  square  cm.  across  the  plane,  and  so  on. 
The  total  energy  transferred  per  square  cm.  per  second 
across  the  plane,  or  the  heat  conductivity  C,  is  therefore 
given  by 

ri_NrSml2nfnd?n+2n'rJ?n+  .  . 

"  2tr    (  NT 

or 


where  d2r  denotes  the  mean  of  the  squares  of  the  displace- 
ments of  the  molecules  in  a  cubic  cm.  If  Sor  denote  the  in- 
ternal specific  heat  at  constant  pressure  per  gram  we  have 
Smr=  Syr^ar,  where  mar  denotes  the  absolute  molecular 
weight  of  a  molecule  r,  and  the  foregoing  equation  may 
therefore  be  written 

d2r  . 

......   (    } 


In  the  case  of  a  mixture  of  molecules  r  and  e  we  have 

r  _  NrmarSgr  d2r  ,  NemafSoe  d2e 

~~"          ~"~" 


It  can  similarly  be  shown  that  if  the  displacement  dr 
is  kept  constant  we  have 

C  =  Nrm2rSard2Tt^,  (227) 


THE  ADMISSIBLE  VALUES  OF  DISPLACEMENTS    261 
and 


where  t.  —1  denotes  the  mean  of  the  reciprocals  of  the  periods 
of  the  molecules  in  a  cubic  cm.  corresponding  to  the  dis- 
placement dTl  and  te~l  has  a  similar  meaning. 

The  least  admissible  value  of  the  average  displacement 
dT  in  the  foregoing  equations  can  be  shown,  similarly  as 
in  the  preceding  Sections,  to  be  the  value  which  satisfies 
the  equation 

4dr  =  lcr, 

where  lcr  denotes  the  heat  transfer  distance  as  defined  in 
Section  37.  The  magnitude  that  may  be  given  to  the  value 
of  dr  is  limited  somewhat  by  the  fact  that  in  the  deduction 
of  the  equations  we  have  assumed  that  the  specific  heat 
along  the  heat  gradient  at  two  points  is  independent  of  their 
distance  apart.  But  this  cannot  hold  unless  the  heat 
gradient  is  taken  infinitely  small.  The  formulae  for  the 
heat  conductivity  may  be  given  forms  taking  the  variation 
of  Smr  and  Nr  along  the  heat  gradient  into  account,  but 
on  account  of  their  complexity  they  possess  no  particular 
interest  or  importance. 

If  the  periods  are  taken  the  same  in  equations  (225) 
and  (215)  and  eliminated  we  obtain  the  equation 


(229) 


where  the  suffixes  c  and  r?  have  been  added  to  the  dis- 
placements to  indicate  to  what  they  refer.  The  right-hand 
side  of  the  equation  is  evidently  in  general  not  equal  to 
unity,  and  hence  the  nature  of  the  motion  of  a  molecule  is 
influenced  in  different  ways  by  a  heat  gradient  and  a  shear- 
ing motion  in  a  substance.  It  should  be  noted  that  cP^ 


262    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

refers  to  unit  heat  gradient,  and  d2^  to  unit  velocity  gradient 
of  the  substance.  According  to  Section  38  the  right-hand 
side  of  the  foregoing  equation  is  greater  than  unity  when 
the  substance  is  in  the  gaseous  state,  and  smaller  than 
unity  when  in  the  liquid  state.  Thus  the  displacement 
of  a  molecule  for  a  given  period  in  a  heat  gradient  is  rela- 
tively greater  than  in  a  velocity  gradient  of  a  substance 
when  it  is  gaseous,  while  the  opposite  holds  when  it  is  in  the 
liquid  state.  This  appears  to  indicate  that  the  path  of  a 
particle  in  a  gas  is  rendered  relatively  more  undulatory 
in  character  by  a  velocity  gradient  than  by  a  heat  gradient, 
while  the  opposite  holds  in  the  case  of  a  liquid. 

The  displacements  and  periods  may  be  expressed  in 
terms  of  more  fundamental  quantities  similarly  as  in  the 
previous  Section.  The  results  may  be  used  to  find  approx- 
imate values  of  the  former  quantities  in  the  case  of  a  mixture 
by  the  help  of  Section  38. 

60.  Another  Method  of  Determining  the  Total 
Average  Velocity  of  Translation  of  a  Colloidal 
Particle. 

We  have  seen  in  Section  56  that  equation  (201)  holds 
only  when  the  period  tT  is  equal  to,  or  larger,  than  that 
corresponding  to  the  average  displacement  which  satisfies 
equation  (202).  The  smallest  value  of  tr  admissible  may 
be  determined  by  using  successively  values  of  tr  in  decreas- 
ing order  of  magnitude  and  observing  the  corresponding 
values  of  the  displacements,  till  values  are  obtained  which 
do  not  satisfy  equation  (201),  or  equation  (206)  if  Stokes' 
law  holds.  The  value  of  tT  corresponding  to  the  transition 
point  is  the  smallest  admissible  value.  On  substituting 
this  value  and  the  value  of  the  corresponding  average  dis- 
placement in  equation  (203)  the  value  obtained  for  Vtr 
is  the  total  average  velocity  of  the  particle. 


TEST  OF  ADMISSIBILITY  OF  DISPLACEMENTS      263 

Since  dr  on  the  average  is  proportional  to  tr  when  its 
value  is  inadmissible  according  to  the  Section  cited,  it  fol- 
lows from  equations  (201)  and  (206)  that  in  that  case 


(A) 


and 


&  r<RT 


37T7Y 


Thus  as  the  value  of  the  period  is  decreased  a  point  is  ulti- 
mately reached  when  the  ratio  d2r/tr  ceases  to  be  constant, 
and  decreases  proportionally  to  tr  for  further  decreases 
in  lr.  Therefore  on  plotting  d2T/tr  against  trj  and  drawing  a 
mean  straight  line  through  the  points  which  indicate  that 
the  ratio  is  constant,  and  a  mean  straight  line  through  the 
other  points,  the  intersection  •  of  the  two  lines  gives  the 
period  t'r  and  displacement  d'r  which  on  being  substituted 
in  equation  (203)  give  the  total  average  velocity  of  the 
colloidal  particle. 

No  experiments  have  of  course  yet  been  carried  out  with 
the  ob'ect  of  determining  the  total  average  velocity  of  a 
coll;  i  lal  ^article  in  this  way.  It  seems  probable,  however, 
that  Nordlund  in  testing  the  constancy  of  d2r/tr  in  the 
experiments  described  in  Section  56  advanced  into  a  region 
in  which  the  values  of  d2r  and  t,  are  inadmissible  in  equation 
(206).  Thus  an  inspection  of  Table  XXIX  shows  that  the 
ratio  in  one  set  of  experiments,  though  constant  for  periods 
of  3£  and  101  inclusive  where  t=  1.481  sec.,  decreases  con- 
siderably for  smaller  periods.  If  this  deviation  is  genuine, 
as  it  seems  to  be,  it  will  be  possible  to  determine  from 
Nordlund's  experiments  whether  the  velocity  of  a  colloidal 
particle  is  the  same  as  if  it  were  in  the  perfectly  gaseous 
state.  The  radius  of  the  mercury  particle  corresponding 
to  the  values  in  the  Table  cited  was  2.66  X10~5  cm.  The 


•264    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

velocity  that  it  would  have  if  it  behaved  as  if  it  were  in  the 
perfectly  gaseous  state  is  3.33X10"1  cm. /sec.  The  average 
value  of  d2r  obtained  by  experiment  corresponding  to 
periods  which  were  multiples  of  1.481  sec.  was  1.813X10"8 
cm.,  and  tr  is  therefore  equal  to  1.481  sec.  The  average 
displacement  is  probably  very  approximately  given  by  the 
square  root  of  d?r,  and  hence  the  apparent  velocity  of  the 
particle  according  to  equation  (203)  is 

3.64XlO-4cm./sec. 

If  this  represents  approximately  the  total  average  velocity 
of  the  particle,  as  is  rendered  probable  on  account  of  the 
deviations  mentioned,  it  follows  that  this  velocity  is  about 
1/1000  of  that  the  particle  would  have  in  the  perfectly 
gaseous  state.  This  result  fits  in  with  those  of  Section  49. 
Some  experiments  by  Henri*  should  be  mentioned  in 
this  connection.  He  obtained  cinematograph  photographs 
of  the  Brownian  motion  of  colloidal  particles  of  rubber  of 
IM  radius  in  water.  The  displacements  observed,  which 
corresponded  to  a  period  of  ^V  of  a  second,  were  four  times 
smaller  than  those  calculated  from  Einstein's  equation. 
This  deviation  might  have  been  caused  by  the  period  being 
smaller  than  the  least  admissible  value  of  the  period,  since 
the  deviations  are  in  the  right  direction.  It  is  true  that 
Henri  tested  whether  WT/tT  is  constant,  though  this  point 
was  not  specially  investigated,  and  found  the  ratio  approxi- 
mately constant.  But  the  deviation  of  this  ratio  from  a 
constant  when  it  occurs  is  probably  very  gradual,  and 
may  therefore  be  approximately  constant  in  the  inadmissible 
region  for  values  lying  between  limits  of  considerable  mag- 
nitude. It  is  difficult  to  see  how  in  Henri's  experiments 
an  error  could  have  come  in  to  produce  a  deviation  of  the 
magnitude  mentioned. 

*  Compt.  Rend.,  146,  1024-1026,  (1908). 


MOLECULAR  VELOCITY  AND  ITS  PROJECTION    265 

It  is  highly  desirable  that  experiments  be  carried  out 
with  the  object  of  evaluating  the  ratio  d2r/tr  of  colloidal 
particles  for  as  small  periods  as  possible  in  order  to  obtain 
in  one  or  more  cases  with  certainty  the  smallest  period  that 
will  satisfy  equation  (206),  and  to  calculate  from  that  the 
total  average  velocity  of  the  particle  concerned.  It  is 
probably  not  accidental  that  in  all  cases  that  have  come 
under  my  notice  the  above  ratio  when  not  fitting  in  with 
equation  (206)  fitted  in  with  the  inequality  (A)  which 
corresponds  to  periods  not  admissible  in  equation  (206). 

61.  The  Distribution  of  the  Molecular  Velocities 
in  a  Substance  not  Obeying  the  Gas  Laws. 

It  does  not  seem  possible  as  yet  to  obtain  any  definite 
theoretical  information  about  this  distribution.  The  proof 
of  Maxwell's  law  given  in  Section  7  applies  only  to  gases, 
and  the  law  need  not  therefore  hold  for  a  substance  not 
obeying  the  gas  laws.  It  is  very  probable,  however,  that 
the  distribution  of  molecular  velocities  obeys  Maxwell's 
law  in  all  cases.  This  may  be  tested  directly  for  colloidal 
particles  in  solution.  Thus  according  to  Section  7  the 
probability  that  the  velocity  of  a  molecule  has  values  lying 
between  Vi  and  V  \-\-dV  \  and  is 


This  expression  may  be  converted  into  one  involving  dis- 
placements (Section  56)  as  follows:  The  projection  of  the 
path  of  a  particle  between  two  points  on  to  an  axis  is  the 
same  as  if  the  points  were  joined  by  a  straight  line.  Now 
if  the  path  of  each  particle  were  divided  into  parts  corre- 
sponding to  the  same  period  tr,  these  joined  by  straight 
lines,  and  the  particle  supposed  to  move  along  them,  the  law 


266    MISCELLANEOUS  APPLICATIONS,  CONNECTIONS 

of  distribution  of  velocities  would  probably  not  be  altered 
if  tr  is  below  a  certain  limit,  where  FI  and  Vp  would  now 
refer  to  the  new  conditions.  Instead  of  directly  transform- 
ing the  foregoing  expression  into  one  involving  displace- 
ments, it  will  be  more  convenient  to  derive  the  required 
expression  from  one  used  in  the  deduction  of  the  foregoing 
probability  expression.  Thus  we  have  seen  in  Section  7 
that  the  probability  that  a  molecule  has  a  component 
velocity  lying  between  a  and  a + da  is 


The  quantity  Vp  in  the  expression  may  be  expressed  in  terms 
of  the  average  dr  of  the  displacements  corresponding  to  the 
period  tr.  According  to  Section  7  we  have 

V 

17  _  o  v  P 

v  t  —  4—^-, 

Vir 

where  Vt,  the  total  average  velocity  of  a  molecule  in  any 
state,  now  takes  the  place  of  Fa,  the  average  velocity  in  the 
gaseous  state.  On  applying  equation  2p  =  ls  in  Section  44 
to  this  case  by  considering  the  different  molecular  velocities 
individually  and  adding  up  the  results,  we  obtain 


The  foregoing  two  equations  give  the  equation  Vp  =  2  VV  dr/tr, 
which  expresses  Vp  in  terms  of  dr.     Therefore   on   substi- 
tuting this  expression  for  Vp,  and  writing  dr/tr  for  a,  in  the 
foregoing  probability  expression,  and  considering  a  numb^ 
of  displacements  Nd,  we  obtain 


VELOCITY  DISTRIBUTION   AND  DENSITY         267 

for  the  number  of  the  displacements  Nd  which  have  values 
lying  between  dr  and  d-\-d(dr)  corresponding  to  the  period  tr. 
The  values  of  the  observed  displacements  appear  to  agree 
with  the  foregoing  law  of  distribution,  and  Maxwell's  law 
for  the  distribution  of  molecular  velocities  holds  therefore,  at 
least  approximately  for  colloidal  particles. 

The  most  probable  displacement,  which  is  obtained  by 
dividing  the  foregoing  expression  by  Nd,  differentiating  it 
with  respect  to  dr,  and  equating  the  result  to  zero,  is  equal  to 
zero.  The  reason  for  this  is  not  difficult  to  see.  Consider 
the  molecular  paths  of  the  same  length  per  period  tr  passing 
in  all  directions  through  any  given  point.  It  is  evident  then 
that  the  number  of  molecules  associated  with  a  given  pro- 
jection increases  with  a  decrease  in  its  value.  It  is  of  impor- 
tance to  point  out  that  the  average  displacement  obtained 
by  direct  observation  will  therefore  tend  to  be  rather  too 
large  than  too  small.  The  total  average  velocity  of  a  col- 
loidal deduced  according  to  Section  56  will  therefore  tend 
to  be  too  large  rather  than  too  small,  a  point  which  is  of 
importance  in  determining  whether  this  velocity  is  the  same 
as  that  corresponding  to  the  gaseous  state. 

The  value  of  the  most  probable  velocity  Vp  in  Maxwell's 
distribution  law  will  depend,  according  to  what  has  gone 
before,  on  the  solvent  in  which  the  particles  are  suspended 
besides  on  their  mass,  and  in  the  case  of  a  molecule  the  value 
of  Vp  will  increase  with  increase  of  density  of  the  substance 
at  constant  temperature,  since  we  have  seen  that  its  total 
average  velocity  increases  with  the  density. 


SUBJECT   INDEX 


Absolute  temperature:    Deduction  of,  12;    correction  of  scale  of,    14; 

zero  of,  14. 

Amplitude:  Of  motion  of  colloidal  particle,  9,  189,  190. 
Atom:     Absolute  mass  of  hydrogen,  7. 
Attraction:     Forces  of,  surrounding  molecules  and  atoms,  46,  47. 

Brownian  motion:  Of  particles  in  liquids,  9,  189,  190,  217-219,  250, 
251;  of  particles  in  gases,  10,  251. 

Calorie:     Definition  of,  38. 

Colloidal  particles:     Preparation  of,  10. 

Conduction  of  heat:  Coefficient  of,  148;  characteristic  function  of, 
154,  159,  161;  displacement  formulae  for,  259-262;  extended  forms 
of  formulae  for,  238-241;  formula  for,  of  pure  gas,  157;  formula 
for,  of  gaseous  mixture,  165;  general  formulae  for,  of  pure  sub- 
stance, 155;  general  formulae  for,  of  mixtures,  164;  interference 
function  of,  154,  158,  162,  163;  mean  transfer  distance  in,  148- 
150;  nature  of  147;  of  gaseous  mixtures,  165;  of  liquids,  162,  163; 
of  pure  gases,  158,  159;  of  solids,  166,  167,  184-186. 

Crookes'  radiometer:     Action  of,  10. 

Diffusion:  Cause  of,  168;  characteristic  function  of,  174,  177,  179- 
181;  coefficient  of,  169;  connection  of  coefficient  of,  with  mobility, 
198;  displacement  formulae  for,  242-249,  252-254;  formula  for,  of 
gases,  179;  general  formula  for,  173;  interference  function  of ,  174, 
175,  178;  Maxwell's  expression  for,  of  gases,  187;  mean  path  of 
colloidal  particle,  192,  193;  tables  of  various  coefficients  of,  180, 199. 

Displacement  and  its  period:  Constancy  of  ratio  of  square  of  first 
to  second,  251;  table  of  values  of,  251. 

Effect:     Joule-Thomson,  46,  47. 

Electric  conductivity:  Formulae  for,  182-186;  ratio  of,  to  heat  con- 
ductivity, 184-186. 

269 


270  SUBJECT  INDEX 

• 

Electrons:     Charge  carried  by  7;  concentration  of,  in  metals,  182,  183; 

dissociation  of  atoms  into,  166;  number  of,  crossing  a  square  cm. 

of  a'  substance  per  second,   167;    variation  of  concentration  of, 

with  change  of  temperature,  1&3. 
Energy:     Equipartition  of  molecular  kinetic,  in  gaseous  mixture,  31; 

internal  molecular,  41,  103;    kinetic,  per  molecule  of  a  gas,  29; 

potential,  of  attraction,  103;  total  internal,  of  a  substance,  103. 

Factor:     For  converting  calories  into  ergs,  38. 

Gases:  Equation  of  pure,  12;  equation  of  mixture  of,  13;  equation 
of,  from  dynamical  considerations,  29,  30;  value  of  R  in  equation 
of,  14. 

Heat:     Latent  of  evaporation,  46,  205. 

Heat  gradient:     Effect  of,  on  molecular  path,  240,  241. 

Interaction :     Of  molecules  and  atoms,  48-50. 

Intrinsic  pressure :     Cause  of  69;   comparison  with  other  quantities, 

73;  formulae  for,  70,  71,  104;   from  equation  of  state,  84;   partial 

intrinsic,  202-206;  tables  of  values  of,  72,  139. 

Linear  diameter  law,  79. 

Matter:    Atomic  nature  of,  3-5. 

Maxwell's  law  of  distribution  of  molecular  velocities:  Consideration 
of,  20-27. 

Molecules:  Formulae  for  number  of,  per  c.c.  in  a  gas,  33,  34;  nature  of 
motion  of,  225,  226;  number  of,  per  gram  molecule,  from  charge 
on  the  electron  and  electrolysis,  7;  from  observed  displacements 
and  Stokes'  law,  250;  from  counting  and  Van't  Hoff's  law,  217-219. 

Mobility:  Connection  of  coefficient  of,  with  osmotic  pressure,  197; 
coefficient  of,  expressed  in  terms  of  molecular  displacements,  249- 
254;  of  a  colloidal  particle  in  an  electric  field,  200,  201;  of  a 
colloidal  particle  in  terms  of  its  path  and  period,  199,  200;  table 
of  coefficients  of,  199. 

Number  of  molecules  n  crossing  a  square  cm.  in  one  direction:  De- 
termination of,  by  an  equation  of  equilibrium,  95-105;  formula 
for,  corresponding  to  any  state,  56;  in  a  gas,  34-37;  inferior  and 
superior  limits  of,  74,  75;  inferior  limit,  of  80;  probability  series 
for,  96. 


SUBJECT  INDEX  271 

Osmotic  pressure:  Cause  of,  in  heterogeneous  mixtures,  208,  209; 
connection  of,  with  molecular  motion  in  a  dilute  solution,  213-217; 
constants  of,  in  heterogeneous  dilute  solution,  217;  in  connection 
with  diffusion,  197;  permanent  nature  of,  21. 

Path  satisfying  given  conditions:     Mean  and  mean  of  squares  of,  109; 

probability  of,  lying  between  given  limits,   107;    probability  of, 

cutting  a  plane,  108. 
Polymerization:     Of  mercury,  105. 
Pressure:     External,  given  by  equation  of  state,  81-86. 
Pressure  of  expansion:     Average,   per  single   molecule  in  any  state, 

35-37;  nature  of,  in  a  substance  in  any  state,  62-64;  of  mixtures, 

67,  68;  total,  in  terms  of  other  quantities,  65,  66. 

Repulsion:     Forces  of,  surrounding  atoms  and  molecules,  47. 

Shearing  motion:     In  viscosity,  110,  111;  effect  of,  on  path  of  particle, 

232. 

Sound:     Formula  for  velocity  of,  41. 
Specific  heats:     Of  gases,  39,  40;    of  liquids  and  dense  gases,  42-44, 

103,  104;  ratio  of  the  two,  of  gases,  40,  41. 
State,  corresponding:     Nature  of,  91-94. 
State,  equations  of:     Conditions  that  they  have  to  satisfy,  at  critical 

point,  88;  at  absolute  zero,  89;  thermodynamical,  90,  91;  various 

forms,  81-86. 
Stokes'  law:     Formulae  involving,  220,  250;  nature  of,  220. 

Thermodynamics:     First  law  of,  37,  38. 

Volume,  external  molecular:     Definition  of,  172,  173. 

Volume,  internal  molecular,  or  b:  Cause  of,  78,  79;  determination  of, 
95-105;  mathematical  definition  of,  58;  properties  of,  58,  59,  61; 
superior  limit  of,  78,  79. 

Volume,  real  and  apparent :     Discussion  of,  77,  78. 

Velocity  of  a  molecule:  Average  kinetic  energy,  in  a  gas,  28;  average, 
in  a  gas,  20;  determination  of  total  average,  95-103,  139;  inferior 
limit  of  total  average,  80;  most  probable,  in  a  gas,  20;  variation 
of,  with  time  in  a  gas,  20;  variation  of,  with  time  in  a  substance  in 
any  state,  265-267;  when  not  under  the  action  of  a  force  in  a  gas 
or  liquid,  52,  53. 

Velocity  of  colloidal  particle:     Determination  of,  195,  210,  262-265. 


272  SUBJECT  INDEX 

. 

Viscosity:  Cause  of,  110,  111;  characteristic  function  of,  121,  128, 
129,  130,  144,  145;  definition  of  coefficient  of,  111;  definition  of 
momentum  transfer  distance  of,  113-116;  displacement  formula) 
for,  255-258;  extended  formulae  for,  230-238;  formula  for,  of  a 
gas,  125,  126,  131;  general  formulae  for,  122,  142,  143;  interfer- 
ence function  of,  121,  134,  138,  140,  141,  145-147;  measurement 
of  coefficient  of,  112;  various  factors  influencing  it,  135-136. 


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